2014 volume 22 no. 1

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2014 Volume 22 No. 1 EDITOR-IN-CHIEF Daniel Florin SOFONEA ASSOCIATE EDITOR Ana Maria ACU HONORARY EDITOR Dumitru ACU EDITORIAL BOARD Heinrich Begehr Shigeyoshi Owa Piergiulio Corsini Detlef H. Mache Aldo Peretti Andrei Duma Vijay Gupta Dorin Andrica Claudiu Kifor Heiner Gonska Dumitru Gaspar Malvina Baica Vasile Berinde Adrian Petrusel SCIENTIFIC SECRETARY Emil C. POPA Nicusor MINCULETE Ioan Tincu EDITORIAL OFFICE DEPARTMENT OF MATHEMATICS AND INFORMATICS GENERAL MATHEMATICS Str. Dr. Ion Ratiu, no. 5-7 550012 - Sibiu, ROMANIA Electronical version: http://depmath.ulbsibiu.ro/genmath/

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Page 1: 2014 Volume 22 No. 1

2014 Volume 22 No. 1

EDITOR-IN-CHIEF

Daniel Florin SOFONEA

ASSOCIATE EDITOR

Ana Maria ACU

HONORARY EDITOR

Dumitru ACU

EDITORIAL BOARD

Heinrich Begehr Shigeyoshi Owa Piergiulio Corsini Detlef H. Mache Aldo Peretti

Andrei Duma Vijay Gupta Dorin Andrica Claudiu Kifor

Heiner Gonska Dumitru Gaspar Malvina Baica Vasile Berinde Adrian Petrusel

SCIENTIFIC SECRETARY

Emil C. POPA Nicusor MINCULETE

Ioan Tincu

EDITORIAL OFFICE

DEPARTMENT OF MATHEMATICS AND INFORMATICS

GENERAL MATHEMATICS

Str. Dr. Ion Ratiu, no. 5-7

550012 - Sibiu, ROMANIA

Electronical version: http://depmath.ulbsibiu.ro/genmath/

Page 2: 2014 Volume 22 No. 1

Contents

V. A. Radu, Academician Professor D.D. Stancu - a life time dedicated to numerical

analysis and theory of approximation .…………………………………………..……………... 3

A.Vernescu, Academician Professor Dimitrie D. Stancu, a respectful remember

and a deep homage ….………………….……………………………………………………..13

A. M. Acu, H. Gonska, On Bullen's and related inequalities ………………………………...19

T. Acar, A. Aral, I. Raşa, Modified Bernstein-Durrmeyer operators………………………...27

A. Florea, E. Păltănea, Some Hermite-Hadamard inequalities for convex functions

on the co-ordinates ……………..……………………………...………………..……………...43

M.M. Birou, A Bernstein-Durrmeyer operator which preserves e0 and e2………………..……49

D. Inoan, I. Raşa, A de Casteljau type algorithm in matrix form …………………………….59

N. Minculete, P. Dicu, A. Raţiu, Two reverse inequalities of Bullen's inequality…………...69

V.G. Cristea, The volume of the unit ball. A review …………………..……………………...75

S. Dumitrescu, Ramanujan type formulas for approximating the gamma function.

A survey ………………………………………………………………………………...……..85

R. Păltănea, Approximation of fractional derivatives by Bernstein operators…………….….91

G. Stan, Uniform approximation of functions by Baskakov-Kantorovich operators………….99

D.I. Duca, E.-L. Pop, Properties of the intermediate point from a mean-value theorem…….109

I. Ţincu, Characterization theorems of Jacobi and Laguerre polynomials…………………...119

A. Baboş, Interpolation operators on a triangle with one curved side……………………..…125

E.C. Popa, Some inequalities for the Landau constants………………………………………133

F. Sofonea, A. M. Acu, A. Rafiq, D. Barbosu, Error bounds for a class of quadrature

formulas……………………………………………………………………………………….139

Page 3: 2014 Volume 22 No. 1

General Mathematics Vol. 22, No. 1 (2014), 3–11

Academician Professor D.D. Stancu - a life timededicated to numerical analysis and theory of

approximation 1

Voichita Adriana Radu

Dedicated to the late Academician Professor Dr. Dimitrie D. Stancu

Abstract

This article wants to be a tribute dedicated to the academician professorDimitrie D. Stancu. His life (1927− 2014) was a beautiful but a hard life, fullof work, honor and success. This article reflects the author personal point ofview and is far from complete, but it is a prospective through the eyes of oneof his last PhD students.

2010 Mathematics Subject Classification: 01A65, 01A70, 41A36.Key words and phrases: Approximation by positive linear operators, Stancu

operators.

1 Academician Professor D.D. Stancu

This spring, on April 17, the mathematical community suffered a big loss: thedecease of Academician Professor D.D. Stancu, a Romanian distinguish mathemati-cian.

1Received 2 July, 2014Accepted for publication (in revised form) 6 August, 2014

3

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4 V.A. Radu

He was an Emeritus member of American Mathematical Society and an Honorarymember of the Romanian Academy. He was also a member of the German societyGesellschaft fur Angewandte Mathematik und Mechanik.

Professor D.D. Stancu was born on February 11, 1927, in a farmer family, fromthe township Calacea, situated not far from Timisoara, the capital of Banat, a south-west province of Romania. In his shoolage he had many difficulties being orphanand very poor, but with the help of his mathematics teachers he succeeded to makeprogress in studies at the prestigious Liceum Moise Nicoara from the large city Arad.

In the period 1947-1951 he studied at the Faculty of Mathematics of the Univer-sity Victor Babes, from Cluj, Romania. When he was a student he was under theinfluence of professor Tiberiu Popoviciu (1906-1975), a great master of NumericalAnalysis and Approximation Theory. He stimulated him to do research work (see[3]).

The main contributions of research work of Professor D.D. Stancu fall into thefollowing list of topics: Approximation of functions by means of linear and posi-tive operators, Representation of remainders in linear approximation procedures,Probabilistic methods for construction and investigation of linear positive opera-tors, Interpolation theory, Spline approximation, Numerical differentiation, Ortho-gonal polynomials, Numerical quadratures and cubatures, Taylor-type expansions,Use of Interpolation and Calculus of finite differences in Probability theory andMathematical statistics.

He has obtained the Ph.D. in Mathematics in 1956 and his scientific advisor forthe doctoral dissertation was professor Tiberiu Popoviciu.

From 1951 he had a continuous academic career at the Babes-Bolyai Univer-sity of Cluj. At the university, professor D.D. Stancu has taught several courses:Mathematical Analysis, Numerical Analysis, Approximation Theory, Informatics,Probability Theory and Constructive Theory of Functions.

From 1968 (in 46 years) he had 46 PhD students from Romania, Germany andVietnam. In the following, we will give the list of the mathe- maticians who’s PhDthesis was conducted by Academician Professor D.D. Stancu, in cronological order.

From 1970 to 1979 :

1 Gligor Moldovan− 19712 Stefan Maruster− 19753 Ioan Gansca− 19754 Alexandru Lupas− 19765 Ion Mihoc− 19766 Trung Du Hoang− 19767 Stefan Cobzas− 19788 Maria Micula− 19789 Octavian Dogaru− 1979

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Academician Professor D.D. Stancu... 5

From 1980 to 1989 :

1 Dumitru Acu− 19802 Dumitru Adam− 19803 Aurel Gaidici− 19804 Horst Kramer− 19805 Maria Mihoc− 19816 Ioan Gavrea− 19827 Petru Blaga− 19838 Adrian Diaconu− 19839 Craciun Iancu− 198310 Ioan Valeriu Serb− 198311 Floare Elvira Kramer− 198412 Constantin Manole− 198413 Traian Augustin Muresan− 198414 Zoltan Kasa− 198515 Leon Tambulea− 198516 Cristina Sanda Cismasiu− 198617 Tiberiu Vladislav− 198618 Maria Dumitrescu− 198919 Teodor Toadere− 1989

From 1990 to 1999 :

1 Dumitru Dumitrescu− 19902 Ioana Chiorean− 19943 Octavian Agratini− 19954 Alexandra Ana Ciupa− 19955 Reiner Dunnbeil− 19966 Alexandru Dan Barbosu− 19977 Gabriela Vlaic− 19988 Emil Catinas− 19999 Daria Elena Dumitras− 199910 Silvia Toader− 1999

From 2000 to 2010 :

1 Andrei Vernescu − 20002 Lucia Cabulea− 20023 Ioana Marcela Gorduza (Tascu)− 20044 Daniel Marius Vladislav− 20045 Maria Craciun− 20056 Voichita Adriana Cleciu (Radu)− 20067 Alina Ofelia Beian (Putura)− 20078 Elena Iulia Stoica (Laze)− 2010

The mathematical legacy of professor D.D. Stancu is spread all over Romaniaand even more, as we can see in the following distribution:

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6 V.A. Radu

2 An attempt to portray D.D. Stancu

In 2002, professor Octavian Agratini (see[1]), using the following sets

• P1=caring father, youth protector

• P2=gifted teacher, minute & skilled tutor, wise & witty speaker

• P3=good organizator, honest judge, ambitious & challenging partner

define the space

DD := P1 ∪ P2 ∪ P3.

Then, he give us the following result:

Theorem 1 We have

DD=sp internationally apreciated mathematician, great heart.

Now, we try to give a natural consequence of O. Agratini’s Theorem.

Let us consider the following :

• PhD=dedicated & serious hard workers PhD students

• f= function of knowledge

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Academician Professor D.D. Stancu... 7

• ||f || = sup|f(x)|, x ∈ PhD

• B(PhD) = f : PhD → R| f bounded by time

• C(PhD) = f : PhD → R| f continuous for a lifetime

Then, we have

CB(PhD) := C(PhD) ∩B(PhD).

Theorem 2 (CB(PhD), ∥ · ∥) is a subspace of DD space.

And if we have the set

F=trusted, challenging & dearest friends , PhD ⊂ F

then

Theorem 3 (CB(F), ∥ · ∥) is a DD space.

3 Stancu operators

In 1968, professor D.D. Stancu introduced and studied a new sequence of linear andpositive operators, constructed using Polya-Markov scheme

Sαn : C[0, 1] → C[0, 1],

(1) (Sαnf) (x) =

n∑k=0

ωα(n,k)(x)f

(k

n

)where

(2) ωα(n,k)(x) =

(n

k

)x[k,−α](1− x)[n−k,−α]

1[n,−α],

n ∈ N and α a real parameter depending only on n (see [13], [16], [7], [2] and [17]).First, we recall the construction of the operator.Let U be an urn with a white balls & b black balls, N := a+ b. A trial consists

in taking a ball from the urn, recording its colour and replacing the ball into theurn, together with another c balls with the same color.

We want to determine the probability that in n repeated trials to get k whiteballs.

We denote by Xkn the desired event. We have to compute P (Xk

n).Also, Wj will be the event of getting a white ball at the jth trial, j = 1, n.The desired event can be written as

Xkn =

∪Wi1 ∩Wi2 ∩ . . . ∩Wik ∩Wik+1

∩ . . . ∩Win

where i1, i2, . . . , in = 1, 2, . . . , n.

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8 V.A. Radu

The previous union contains(nk

)terms since we can obtain k white balls at any

k trials from n trials.

The desired probability is :

P (Xkn) = P

(∪Wi1 ∩Wi2 ∩ . . . ∩Wik ∩Wik+1

∩ . . . ∩Win)

=∑

P (Wi1) · P (Wi2) . . . P (Wik) · P (Wik+1) . . . P (Win)

=∑

pi1 · pi2 . . . pik · qik+1. . . qin

then we have

P (Xkn) =

(n

k

)a(a+ c) . . . (a+ k − 1c)b(b+ c) . . . (b+ n− k − 1c)

N(N + c)(N + 2c) . . . (N + n− 1c).

If we use the following notation aN = x, x ∈ (0, 1), b

N = 1− x, cN = α then we have

the Stancu fundamental polynomials

ωα(n,k)(x) =

(nk

)x(x+α)...(x+k−1α)(1−x)(1−x+α)...(1−x+n−k−1α)

(1+α)(1+2α)...(1+n−1α),

and using the notation of the factorial powers, with natural exponent and real step,we have

ωα(n,k)(x) =

(n

k

)x[k,−α](1− x)[n−k,−α]

1[n,−α].

This initial Stancu operators have been intensivly studied by numerous foreign andRomanian mathematicians. In 1968,1970 and 1971, in [13], [14] and [15] professorD.D. Stancu proved the fallowing properties:

• this operators are linear and positive,

• are interpolating at the ends of the interval [0, 1],

• for α = 0, we find the classical Bernstein operators,

• for α = − 1n , (2) becomes the Lagrange interpolation polynomials,

• the following identities hold true:(Sα

ne0) (x) = 1(Sα

ne1) (x) = x

(Sαne2) (x) =

1nx+ n−1

n · x(x+α)1+α , x ∈ [0, 1]

• the convergence theorem are proved,

• estimations of the rate of convergence in terms of modulus of continuity aregiven,

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Academician Professor D.D. Stancu... 9

• the operators (1) can be represented by means of the Beta function, the Bern-stein operators and by finite and divided differences,

• the norm of the operator is ||Sαn || = 1,

• the operators generate the approximation formula

f(x) = (Sαnf)(x) + (Rα

nf)(x),

• different representations of the remainder are given.

In addition, other mathematicians have completed the list of properties of Stancuoperators.

In 1978 and 1980, in [11] and [12] G. Mastroianni and M. Occorsio disscusedabout the variation diminishing property and about the derivatives of Stancu’s poly-nomials.

In 1984, in [9] H.H. Gonska and J. Meier found better constants for estimationsof the rate of convergence, in terms of modulus of continuity.

In 1988 and 1989, in [5] and [6] B. Della Vechia gave some recurrence formulaand an elementary proof of the preservation of Lipschitz constants by the Stancuoperators.

In 2002, in [10] A. Lupas and L. Lupas proved a representation of the remainderterm and also some mean value theorems. In addition they discussed a quadratureformula for Stancu operators.

Also, in 2002, in [8] Z. Finta present direct and converse results for the operators(1). Moreover, he proved the equivalence of ∥Sα

nf − f∥ and ∥Bnf − f∥.In 2003, in [4] the author revealed the preservation of global smoothness of the

Stancu operators, using the modulus of continuity and K- functionals.

4 Conclusions

This paper tried to gather (in chronological order) the most significant propertiesof the operators discovered and proved either by professor D.D. Stancu or by thosewho have investigated them.

After the pioneer work of professor D.D. Stancu, these operators have been suc-cessfully used by other mathematicians to study properties of linear positive methodsof approximation. There are numerous combinations and generalizations of Stancuoperators: Bernstein-Stancu operators, Schurer-Stancu operators, Durrmeyer-Stancuoperators, Baskakov-Durrmeyer-Stancu operators, Stancu-Hurwitz operators,Kantorovich-Stancu operators and many others.

Professor D.D. Stancu publication lists about 160 items (papers and books). Theintensive research work and the important results obtained by professor D.D. Stancuhas brought to him international recognition and appreciation. Serch for ”Stancuoperators” on the Web of Science, returned more then 150 results as journal articles(there are more than 70 papers containing his name in their titles).

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10 V.A. Radu

We conclude with the words of William Arthur Ward:

”The good teacher explains.The superior teacher demonstrates.

The great teacher inspires.”

References

[1] O. Agratini, An attempt to portray D.D. Stancu, Proceedings of the Interna-tional Symposium on Numerical Analysis and Approximation Theory Dedicatedto the 75th Anniversary of D.D. Stancu, Cluj-Napoca, May 9-11 (2002), 16–18.

[2] O. Agratini, Aproximare prin operatori liniari, Editura Presa Universitara Clu-jeana, Cluj-Napoca, 2000, pp. 354.

[3] P. Blaga, O. Agratini, Academician professor Dimitrie D. Stancu at his 80th

birthday anniversary, Studia Univ.Babes-Bolyai, Mathematica, 52 (2007), no.4, 3–7.

[4] V.A. Cleciu, On some classes of Bernstein type operators which preserve theglobal smoothness in the case of univariate functions, Acta Universitatis Apu-lensis Mathematics - Informatics, (2003), 91–100.

[5] B. Della Vechia, On Stancu operator and its generalizations, Inst. Applicazionidella Matematica (Rapp. Tecnico), Napoli, 47 (1988).

[6] B. Della Vechia, On the preservation of Lipschitz constants for some linearoperators, Bol.Un Mat.Ital. B, 3 (1989), no. 7, 125–136.

[7] B. Della Vechia, On the approximation of functions by means of the operators ofD.D. Stancu, Studia Univ.Babes-Bolyai, Mathematica, 37 (1992), no. 1, 3–36.

[8] Z. Finta, Direct and converse results for Stancu operator, Periodica Mathema-tica Hungarica, 44 (2002), no. 1, 1-6.

[9] H.H. Gonska, J. Meier, Quantitative theorems on approximation by Bernstein-Stancu operators, Calcolo, fasc. IV, 21 (1984), 317–335.

[10] A. Lupas, L. Lupas, Properties of Stancu operators, Proceedings of the Inter-national Symposium on Numerical Analysis and Approximation Theory Dedi-cated to the 75th Anniversary of D.D. Stancu, Cluj-Napoca, May 9-11 (2002),258–275.

[11] G. Mastroianni, M. Occorsio, Sulle derivate dei polinomi di Stancu, Rend.Accad. Sci. Fis. Mat. Napoli, serie. IV, 45 (1978), 273–281.

[12] G. Mastroianni, Su una clase di operatori lineari e positivi, Rend. Accad. Sci.Fis. Mat. Napoli, serie. IV, 48 (1980), 217–235.

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Academician Professor D.D. Stancu... 11

[13] D.D. Stancu, Approximation of functions by a new class of linear positiveoperators, Rev.Roum.Math.Pures et Appl., 13 (1968), 1173–1194.

[14] D.D. Stancu, Approximation properties of a class of linear positive operators,Studia Univ.Babes-Bolyai, seria Mathematica-Mechanica, 15 (1970), no. 2,33–38.

[15] D.D. Stancu, On the remainder of approximation of functions by means ofa parameter-dependent linear polynomial operator, Studia Univ.Babes-Bolyai,seria Mathematica-Mechanica, 16 (1971), no. 2, 59–66.

[16] D.D. Stancu, Approximation of functions by means of some new classes ofpositive linear operators, ”Numerische Methoden der Approximationstheorie”,Proc.Conf.Oberwolfach 1971 ISNM vol 16, B. Verlag, Basel (1972), 187–203.

[17] D.D. Stancu, Gh. Coman, O. Agratini, R. Trambitas, Analiza numerica si teoriaaproximarii, vol I, Ed. Presa Univ. Clujeana, Cluj-Napoca, 2001, pp. 414.

Voichita Adriana RaduBabes Bolyai UniversityFaculty of Economics and Business AdministrationDepartment of Statistics-Forecasts-Mathematics58-60 T. Mihali, 400591 Cluj-Napoca, Romaniae-mail: [email protected]

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General Mathematics Vol. 22, No. 1 (2014), 13–17

Academician Professor Dimitrie D. Stancu, a respectfulremember and a deep homage 1

Andrei Vernescu

Dedicated to the late Academician Professor Dr. Dimitrie D. Stancu

Abstract

The presentation consists in a remembering of the life and the work of ourbeloved master Acad. Prof. D. D. Stancu.

2010 Mathematics Subject Classification: 01A60, 41A10, 41A25, 41A36,41A80, 65D25, 65D30.

Key words and phrases: Numerical Analysis, Approximation Theory,Interpolation theory, Numerical differentiation, Orthogonal Polynomials,

Numerical quadratures and cubatures, Taylor-type expansions, Linear positiveoperators of approximation, finite operatorial calculus.

This spring, on April 17, we lost our beloved master and adviser, ProfessorDimitrie D. Stancu. The chief of the Romanian School of Numerical Analysis andApproximation Theory was a very important professor of “Babes-Bolyai” University,an honorary member of the Romanian Academy and a Doctor Honoris Causa of theUniversity “Lucian Blaga” of Sibiu and of the North University of Baia Mare (nowthe North Universitary Center of the Technical University of Cluj-Napoca).

The death of Academician Professor Dimitrie D. Stancu is a painful shock notonly for his family, but also for all his friends, contributors, disciples, former studentsand former Ph. D. students and for the entire mathematical community. We all arevery sad and we remember now not only the great mathematician, the head of theimportant Romanian Mathematical School of Numerical Analysis and Approxima-tion Theory, but also we think to the man, to his pleasant, warm, optimistic andgenerous personality.

Let’s remember some events and facts related to the life and work of ProfessorDimitrie D. Stancu. It constitutes a model of the action against the difficulties of

1Received 29 June, 2014Accepted for publication (in revised form) 20 August, 2014

13

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14 A. Vernescu

the unfavourable ”initial conditions” of the life, to accomplish a beautiful life, careerand mathematical work. Indeed, he was born on February 11, 1927 in a very poorfamily in the village Calacea, situated near Timisoara, the capital of the provinceof Banat, he remained orphan and was forced to work when he was too young.Fortunately, his oldest brother took him in Arad at ”Regina Maria” (”The QueenMary”) orphanage, where there was included in an elementary school. After this, hestudied during the years 1943 and 1947 at the prestigious highschool ”Moise Nicoara”of Arad. This educational institution has already given to the cultural and scientificcommunity several personalities, including two academicians in the domain of ma-thematics: Tiberiu Popoviciu (1906-1975) and Caius Iacob (1912-1992). Dimitrie D.Stancu was a brilliant highschool student and his native talent in mathematics wasnoticed by his teachers and especially by the one in mathematics, Ascaniu Crisan,the school principal. In 1947 Dimitrie D. Stancu began his four-year study in ma-thematics at the ”Babes-Bolyai” University from Cluj (today Cluj-Napoca), a townwith a very important cultural life, a famous universitary center and the capital ofthe historical Romanian province of Transylvania. Immediately the very talentedbut also very sympathetic student was discovered by the great Professor TiberiuPopoviciu. This professor, one of the most important and influential professorsof the University, a great master of the Theory of Approximation, influenced andstimulated the work of research in mathematics of the young Stancu, which wasnamed ”preparator” since his third year of studies. In 1951, after his graduation, hewas engaged assistant at the Department of Mathematical Analysis, of the Universityof Cluj. The pleasant town of Cluj will become the new residence of D. D. Stancu.He obtained the Ph. D. in mathematics in 1956, for which the thesis (ready since1955) was intitled “A study of the polynomial interpolation of functions of severalvariables, with applications to the numerical differentiation and integration; methodsfor evaluating the remainders” (in Romanian) and had 192 pages. His adviser wasTiberiu Popoviciu and in the examination comission were Miron Nicolescu, DumitruV. Ionescu and Adolf Haimovici.

In a normal succession, D. D. Stancu advanced up to the rank of full professorin 1969. He holds a continuous academic career at the University ”Babes-Bolyai”,excepting the year 1961-1962, when he had a fellowship at the Numerical AnalysisDepartment of the University of Wisconsin, Madison, conducted by late professorPreston C. Hammer. Here he spent at the University of Wisconsin during theacademic year 1961-1962 and had direct scientific contacts with important mathe-maticians of the time: J. Favard, A. Ostrovski, I. J. Schoenberg, G. G. Lorentz, A.Cheney, P. L. Butzer, A. Sharma and other. Dimitrie D. Stancu also participatedto several mathematical events organized by the American Mathematical Society atMilwaukee, Chicago and New York.

When he returned in Romania, he was named deputy dean of the faculty ofMathematics of his University and head of the new created Chair of Numerical andStatistical Calculus; he was the head of this chair between 1962 and 1995, when heretired.

He had a very nice family: his wife dr. Felicia Stancu also worked in mathematics

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Acad. Professor Dimitrie D. Stancu ... 15

in the same University. His daughters Angela (1957) and Mirela (1958) are teachingmathematics in highschools of Cluj-Napoca. He has three grandsons: Alexandru-Mircea Scridon (1983) and George Scridon (1992), the sons of Mirela and StefanaMunteanu (1991), the daughter of Angela. For his family and friends, Dimitrie D.Stancu was also named sometimes Didi (from his initiales D. D.)

Professor D. D. Stancu taught Mathematical Analysis, Numerical Analysis,Approximation Theory, Constructive Theory of Functions, Computer Science,Probability Theory and Mathematical Statistics. He published at Babes-BolyaiUniversity in 1977 his course of Numerical Analysis and Approximation Theory andlater he published in cooperation the fundamental treatise [9], [10], [11], a veritable”bible” in the domain.

In the 60’s, having a large vision of the future evolutions, he was one of thepromotors of the use of the computers, as Grigore C. Moisil at Bucharest.

Professor D. D. Stancu had a very rich research work in Interpolation The-ory, Numerical differentiation, Orthogonal Polynomials, Numerical quadratures andcubatures, Taylor-type expansions, Approximation of functions by linear positiveoperators, Representation of remainders in linear approximation formulas, proba-bilistic methods for construction and investigation of linear positive operators ofapproximation, use of interpolation and of calculs of finite differences in probabi-lity theory and mathematical statistics, use of the finite operatorial calculus (um-bral calculus) in the construction of operators of approximation. He introduced afamous approximation operator named today the operator of Stancu. Consequently,many operators may be considered by a point of vue of an operator of Stancu and theoperators of Bernstein-Stancu, Kantorovich-Stancu, Schurer-Stancu, Stancu-Mulbachand other are intensively studied.

An list of selected publications of Professor Dimitrie D. Stancu is given in [1].

Also, note that the name of D. D. Stancu appears in many titles of researchpapers of foreign authors; a list of these until 2002 is given in [7], pp. 10-15. Afterthis year, also other many papers contain in the title the name of D. D. Stancu.

Professor Dimitrie D. Stancu guided many Ph. D. students (Romanians andforeign). A list of the first 40 doctoral students of our professor and of the corres-ponding titles of the thesis is given in [7], pp. 6-9.

He was a member of the American Mathematical Society (since 1961), of the Ger-man Society “Gesellschaft fur Angewandte Mathematik und Mechanik” (GAMM).He was the Editor in Chief of the prestigious journal published by the RomanianAcademy “Revue d’Analyse Numerique et de Theorie de l’Approximation” and manyyears ago he becomes a member of the Editorial Board of the famous Italian journal“Calcolo”, published now by “Springer-Verlag”. He was a member of the edito-rial comitees of “Studia Univ. Babes-Bolyai” and “Mathematica” He has doneextensive reviewing, especially in “Mathematical Reviews” and “Zentralblatt furMathematik”.

Professor D. D. Stancu took part to numerous scientific events: Gattlinburg(TN) in USA, Lancaster and Durham in England, Stuttgart, Hannover, Hamburg,Gottingen, Dortmund, Munchen, Siegen, Wurzburg, Berlin and Oberwolfach in Ger-

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16 A. Vernescu

many, Roma, Napoli, Potenza, L’Aquila in Italy, Budapest, Paris, Sofia and other.

He has been invited to present lectures at several Universities from USA, Ger-many, Holland and other.

Under the leadership of professor D. D. Stancu were organized several scientificevents in Cluj-Napoca: “International Conference on Approximation and Optimiza-tion” (1996), “International Symposium on Numerical Analysis and ApproximationTheory” (2002) and ”International Conference on Numerical Analysis and Appro-ximation Theory” (2006). Professor Stancu also took part at several editions of theRoGer Seminars.

We remember always with respect and gratitude the wonderful personality ofour brilliant mathematician, of our beloved Professor and adviser, DIMITRIE D.STANCU.

References

[1] D. Acu, A. Lupas, Professor dr. Honoris Causa Dimitrie D. Stancu at hisanniversary, General Mathematics Vol. 10, No. 1-2 (2002), 3-20.

[2] O. Agratini, An attempt to portray D. D. Stancu, Proc. of the Int. Symp. onNumerical Analysis and Approximation Theory, Cluj-Napoca, May 9-11, 2002,dedicated to the Anniversary of Professor Dr. Dimitrie D. Stancu, Edited byRadu T. Trambitas, Cluj University Press, 16-18.

[3] G. St. Andonie, Istoria matematicii ın Romania Vol. 3, Editura Stiintifica,Bucuresti, 1967, pp. 320-324.

[4] Gh. Coman, I. Pavaloiu, Academician D. D. Stancu and his eightieth birthdayanniversary, Revue d’Analyse Numerique et de Theorie de l’ApproximationTome 36, No. 1, 2007, 5-8.

[5] Gh. Coman, I. A. Rus, A. Vernescu, Academicianul D. D. Stancu la 75 de ani,Gazeta Matematica Seria A, vol. 20 (49) (2002), 55-56.

[6] A.Vernescu, Aniversarea a 80 de ani ai Academicianului Profesor Dimitrie D.Stancu la Universitatea Babes-Bolyai, Gaz. Mat. Seria A, vol. 25 (54), 2007,248-251.

[7] A.Vernescu, Professor dr. honoris causa Dimitrie D. Stancu, honorary memberof the Romanian Academy, at his birthday, Proc. of the Int. Symp. on NumericalAnalysis and Approximation Theory, Cluj-Napoca, May 9-11, 2002, dedicatedto the Anniversary of Professor Dr. Dimitrie D. Stancu, Edited by Radu T.Trambitas, Cluj University Press.

[8] D. D. Stancu, Curs si Culegere de probleme de Analiza Numerica, vol. 1, Univer-sitatea “Babes-Bolyai”, Cluj-Napoca, 1977.

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Acad. Professor Dimitrie D. Stancu ... 17

[9] D. D. Stancu, Gh. Coman, O. Agratini, R. Trambitas, Analiza numerica siTeoria Aproximarii, vol. I, Presa Universitara Clujeana, Cluj- Napoca, 2001.

[10] D. D. Stancu, Gh. Coman, P. Blaga, Analiza numerica si Teoria Aproximarii,vol. II, Presa Universitara Clujeana, Cluj- Napoca, 2002.

[11] O. Agratini, I. Chiorean, Gh. Coman, R. Trambitas, Analiza numerica si TeoriaAproximarii, vol. III, Presa Universitara Clujeana, Cluj- Napoca, 2002. (coor-donatori D. D. Stancu, Gh. Coman).

Andrei VernescuValahia University of TargovisteFaculty of Sciences and ArtsDepartment of SciencesBd. Unirii 118, Targovistee-mail: [email protected]

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Page 19: 2014 Volume 22 No. 1

General Mathematics Vol. 22, No. 1 (2014), 19–26

On Bullen’s and related inequalities 1

Ana Maria Acu, Heiner Gonska

Dedicated to the late Academician Professor Dr. Dimitrie D. Stancu

Abstract

The estimate in Bullen’s inequality will be extended for continuous functionsusing the second order modulus of smoothness. A different form of this inequa-lity will be given in terms of the least concave majorant. Also, the compositecase of Bullen’s inequality is considered.

2010 Mathematics Subject Classification: 26D15, 26A15.

Key words and phrases: Bullen’s inequality, K-functional, modulus ofcontinuity.

1 Motivation

Over the years it happened during several editions of RoGer - the Romanian-GermanSeminar on Approximation Theory - that the second author learned about inequa-lities the validity of which was known for regular (i.e., differentiable) functions only.Using tools from Approximation Theory, we showed in [1] and [2] that such restric-tion can sometimes be dropped and that the estimates can be extended to (at least)continuous functions on the given compact intervals, for example.

Our more general estimates in [1] and [2] were given in terms of the least concavemajorant of the usual first order modulus of continuity. Already this is rather acomplicated quantity. There we focussed on Ostrowski- and Gruss-type directions.The best way seems to be that via a certain K-functional. This road was recentlyand thoroughly described in a paper by Paltanea [10]. But the knowledge aboutthis method is much older. See papers by Peetre [11], Mitjagin and Semenov [9] aswell as the diploma thesis of Sperling [12], for example.

1Received 15 June, 2014Accepted for publication (in revised form) 15 July, 2014

19

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20 A. M. Acu, H. Gonska

In Sections 2 and 3 we will consider classical and composite Bullen functionals.It will become clear there that it is quite natural to use the second order modulusof smoothness of a continuous function and a related K-functional.

At the end of this note we will return to the concave majorant and its significancein the field of Inequalities. For the composite case and continuously differentiablefunctions Section 4 contains a very precise estimate in terms of the majorant.

2 Bullen’s inequality revisited

This paper is mostly meant to be a contribution to the ever-lasting discussion onan inequality given by Bullen in [3] (see also an earlier paper by Hammer [7]) in thefollowing form:

Theorem A. If f is convex and integrable, then(∫ 1

−1f

)− 2f(0) ≤ f(−1) + f(1)−

∫ 1

−1f.

If transformed to an arbitrary compact interval [a, b] ⊂ R, a < b, the equivalent formof the inequality reads

2

b− a

∫ b

af(x)dx ≤ f(a) + f(b)

2+ f

(a+ b

2

).

This is the form we learned about at ”RoGer 2014 - Sibiu”. In the talk of PetricaDicu we also learned that in 2000 Dragomir and Pearce [4] had given the followinginequality for functions f in C2[a, b] with known bounds for f ′′:

Theorem 1 Let f : [a, b] → R be a twice differentiable function for which thereexist real constants m and M such that

m ≤ f ′′(x) ≤M, for all x ∈ [a, b].

Then

(1)m(b− a)2

24≤ f(a) + f(b)

2+ f

(a+ b

2

)− 2

b− a

∫ b

af(x)dx

≤ M(b− a)2

24.

If we define the Bullen functional B by

B(f) :=f(a) + f(b)

2+ f

(a+ b

2

)− 2

b− a

∫ b

af(x)dx,

we note that B is defined for all functions in C[a, b] and that - so far - we have thefollowing

Page 21: 2014 Volume 22 No. 1

On Bullen’s and related inequalities 21

Proposition 1 The Bullen functional B : C[a, b] → R satisfies

(i) |B(f)| ≤ 4||f ||∞ for all f ∈ C[a, b], || · ||∞ indicating the sup norm on [a, b].

(ii) |B(g)| ≤ (b− a)2

24· ||g′′||∞ for all g ∈ C2[a, b].

We will next explain how to turn this into a more general statement using thefollowing result from [6]:

Theorem 2 Let (F, || · ||)F ) be a Banach space, and let H : C[a, b] → (F, || · ||F ) bean operator, where

a) ||H(f + g)||F ≤ γ(||Hf ||F + ||Hg||F ) for all f, g ∈ C[a, b];

b) ||Hf ||F ≤ α · ||f ||∞ for all f ∈ C[a, b];

c) ||Hg||F ≤ β0 · ||g||∞ + β1 · ||g′||∞ + β2 · ||g′′||∞ for all g ∈ C2[a, b].

Then for all f ∈ C[a, b], 0 < h ≤ (b− a)/2, the following inequality holds:

||Hf ||F ≤ γ

β0 · ||f ||∞+

2β1hω1(f ;h)+

3

4

(α+β0+

2β1h

+2β2h2

)ω2(f ;h)

.

Taking H = B and F = R in Theorem 2 we have the following list of constants:

γ = 1, α = 4, β0 = 0, β1 = 0, β2 =(b− a)2

24.

This takes us to the following

Proposition 2 For the Bullen functional B and all f ∈ C[a, b] we have

|B(f)| ≤

(3 +

(b− a

4h

)2)

· ω2(f, h), 0 < h ≤ b− a

2.

The special choice h =b− a

k, k ≥ 2 yields

|B(f)| ≤(3 +

k2

16

)· ω2

(f,b− a

k

).

Remark 1 For f ∈ C2[a, b] the latter inequality implies

|B(f)| ≤(

1

16+

3

k2

)(b− a)2||f ′′||∞, k ≥ 2.

As far as the constant

(1

16+

3

k2

)is concerned, this is much worse than what was

invested for C2 functions.

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22 A. M. Acu, H. Gonska

An alternative estimate is given in the next proposition. Note that it also followsfrom Theorem 6 in Gavrea’s paper [5].

Proposition 3 If the second K-functional on C[a, b] is defined by

K(f ; t2;C[a, b], C2[a, b]) := inf||f − g||∞ + t2||g′′||∞ : g ∈ C2[a, b], t ≥ 0,

then

|B(f)| ≤ 4K

(f ;

(b− a)2

96;C[a, b], C2[a, b]

).

Proof. In [1] the following result was obtained:∣∣∣∣12 [(x− a)f(a) + (b− a)f(x) + (b− x)f(b)]−∫ b

af(t)dt

∣∣∣∣≤ 2(b− a)K

(f ;

(x− a)3 + (b− x)3

24(b− a);C[a, b], C2[a, b]

).

The proposition is proved if we substitute x =a+ b

2in the above inequality.

Remark 2 (i) For h ∈ C2[a, b], we have

|B(h)| ≤ 4 ·K(h;

(b− a)2

96;C[a, b], C2[a, b]

)= 4 · inf

||h− g||∞ +

(b− a)2

96||g′′||∞, g ∈ C2[a, b]

≤ 4 · (b− a)2

96||h′′||∞ (taking g=h)

=(b− a)2

24||h′′||∞, i.e.,

the original inequality for C2 functions.

(ii) It is known that, for f ∈ C[a, b],

K(f ; t2;C[a, b], C2[a, b]) ≤ c · ω2(f, t), 0 ≤ t ≤ b− a

2

with a constant c = c(f, t). According to our knowledge the best possible valueof c is unknown.

Zhuk showed in [13] that, for t ≤ b− a

2, one has

K(f ; t2;C[a, b], C2[a, b]) ≤ 9

4· ω2(f ; t).

Using the latter we arrive, for h ∈ C2[a, b], at

|B(h)| ≤ 3(b− a)2

32∥h′′∥∞.

Page 23: 2014 Volume 22 No. 1

On Bullen’s and related inequalities 23

3 The composite case

Here we consider the composite case of the Bullen functional, i.e., the functionalwhich arises when comparing the composite trapezoidal and midpoint rules. To thisend the interval [a, b] is divided in n ≥ 1 subintervals as follows

a = x0 < · · · < xi < xi+1 < · · · < xn = b.

Let the composite Bullen functional Bc : C[a, b] → R be given by

Bc(f) =1

b− a

n−1∑i=0

(xi+1 − xi)

[f(xi) + f(xi+1)

2+ f

(xi + xi+1

2

)− 2

xi+1 − xi

∫ xi+1

xi

f(x)dx

].

Proposition 4 In the composite case there holds

(i) |Bc(f)| ≤ 4||f ||∞ for all f ∈ C[a, b],

(ii) |Bc(g)| ≤1

24(b− a)

n−1∑i=0

(xi+1 − xi)3||g′′||∞ for all g ∈ C2[a, b].

Using Theorem 2 and Proposition 4 we obtain the following inequality for the com-posite Bullen functional involving the second modulus of continuity:

Proposition 5 For the composite Bullen functional one has

|Bc(f)| ≤

(3 +

1

16(b− a)h2

n−1∑i=0

(xi+1 − xi)3

)ω2(f, h), 0 < h ≤ b− a

2.

For h =1

k

√∑n−1i=0 (xi+1 − xi)3

b− a, k ≥ 2, this yields

|Bc(f)| ≤(3 +

k2

16

)ω2

f, 1k

√∑n−1i=0 (xi+1 − xi)3

b− a

, k ≥ 2.

Remark 3 For f ∈ C2[a, b] the latter inequality implies

|Bc(f)|≤(

1

16+

3

k2

) ∑n−1i=0 (xi+1−xi)3

b− a∥f ′′∥∞≤

(1

16+

3

k2

)(b− a)2∥f ′′∥∞.

The requirement F (x0, x1, . . . , xn) =

n−1∑i=0

(xi+1−xi)3 → minimum entails xi+1−xi =

b− a

n, i = 0, n− 1.

Page 24: 2014 Volume 22 No. 1

24 A. M. Acu, H. Gonska

The inequality involving a K-functional is given next.

Proposition 6 For the functional Bc given as above, f ∈ C[a, b], we have

(2) |Bc(f)| ≤ 4K

(f ;

1

96(b− a)

n−1∑i=0

(xi+1 − xi)3;C[a, b], C2[a, b]

).

Proof. Let g ∈ C2[a, b] arbitrary. Then, for f ∈ C[a, b],

|Bc(f)| ≤ |Bc(f − g)|+ |Bc(g)|

≤ 4∥f − g∥∞ +1

24(b− a)

n−1∑i=0

(xi+1 − xi)3∥g′′∥∞

= 4

∥f − g∥∞ +

1

96(b− a)

n−1∑i=0

(xi+1 − xi)3∥g′′∥∞

.

Passing to the infimum over g yields inequality (2).

4 Composite Bullen functional for f ∈ C1[a, b]

Using the least concave majorant of the modulus of continuity in this section weconsider Bullen’s inequality for f ∈ C1[a, b].

Proposition 7 If f ∈ C1[a, b], then

|Bc(f)| ≤ ω

(f ′,

1

24(b− a)

n−1∑i=0

(xi+1 − xi)3

).

Proof. We have

|Bc(f)| ≤1

b− a

n−1∑i=0

(xi+1 − xi)

∣∣∣∣f(xi) + f(xi+1)

2+ f

(xi + xi+1

2

)− 2

xi+1 − xi

∫ xi+1

xi

f(x)dx

∣∣∣∣=

1

b− a

n−1∑i=0

(xi+1− xi)

∣∣∣∣f(xi) + f(xi+1)

2− 1

xi+1 − xi

∫ xi+1

xi

f(x)dx

+ f

(xi + xi+1

2

)− 1

xi+1 − xi

∫ xi+1

xi

f(x)dx

∣∣∣∣=

1

b− a

n−1∑i=0

∣∣∣∣∫ xi+1

xi

(f(xi)− f(x)

2+f(xi+1)− f(x)

2

+ f

(xi + xi+1

2

)− f(x)

)dx

∣∣∣∣ ≤ 2∥f ′∥∞.

Page 25: 2014 Volume 22 No. 1

On Bullen’s and related inequalities 25

Let g ∈ C2[a, b]. Using Proposition 4 and the latter inequality implies

|Bc(f)| = |Bc(f − g + g)| ≤ |Bc(f − g)|+ |Bc(g)|

≤ 2∥(f − g)′∥∞ +1

24(b− a)

n−1∑i=0

(xi+1 − xi)3∥g′′∥∞

= 2

∥(f − g)′∥∞ +

1

48(b− a)

n−1∑i=0

(xi+1 − xi)3∥g′′∥∞

.

Passing to the infimum over g ∈ C2[a, b] we have

|Bc(f)| ≤ 2K

(f ′;

1

48(b− a)

n−1∑i=0

(xi+1 − xi)3;C1[a, b], C2[a, b]

),

so the result follows as a consequence of the relation (see [10])

K(f ′; t;C1[a, b], C2[a, b]

)=

1

2ω(f ′, 2t), 0 ≤ t ≤ b− a

2.

A particular consequence of Proposition 7 is the following version of Bullen’sinequality.

Proposition 8 If f ∈ C1[a, b], then

|B(f)| ≤ ω

(f ′,

(b− a)2

24

).

Acknowledgment. The authors most gratefully acknowledge the efficient help ofBirgit Dunkel (University of Duisburg-Essen) and Elsa while preparing this note.

References

[1] A. Acu, H. Gonska, Ostrowski inequalities and moduli of smoothness, ResultsMath. 53 (2009), 217-228.

[2] A. Acu, H. Gonska, I. Rasa, Gruss- and Ostrowski-type inequalities in Appro-ximation Theory, Ukrain. Mat. Zh. 63 (2011), 723-740, and Ukrainian Math. J.63 (2011), 843-864.

[3] P.S. Bullen, Error estimates for some elementary quadrature rules, Univ.Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 602-633 (1978), 97-103(1979).

[4] S.S. Dragomir, Ch.E.M. Pearce, Selected Topics on Hermite-HadamardInequalities and Applications, RGNMIA Monographs, Victoria University, 2002(amended version).

Page 26: 2014 Volume 22 No. 1

26 A. M. Acu, H. Gonska

[5] I. Gavrea, Preservation of Lipschitz constants by linear transformations andglobal smoothness preservation, Functions, Series, Operators (L.Leindler, F.Schipp, J. Szabados, eds.), Budapest, 2002, pp. 261-275.

[6] H. Gonska, R. Kovacheva, The second order modulus revisited: remarks, appli-cations, problems, Confer. Sem. Mat. Univ. Bari No. 257 (1994), 32 pp. (1995).

[7] P.C. Hammer, The midpoint method of numerical integration, Math. Mag. 31(1957/1958), 193–195.

[8] N. Minculete, P. Dicu, A. Ratiu, Two reverse inequalities of Bullen’s inequalityand several applications, Lecture given at RoGer 2014 - Sibiu.

[9] B.S. Mitjagin, M.E. Semenov, Absence of interpolation of linear operators inspaces of smooth functions (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 41(1977), no. 6, 1289–1328, 1447.

[10] R. Paltanea, Representation of the K-functional K(f, C[a, b], C1[a, b], ·) - a newapproach. Bull. Transilv. Univ. Brasov Ser. III 3(52) (2010), 93–99.

[11] J. Peetre, On the connection between the theory of interpolation spaces andapproximation theory. Proc. Conf. Constructive Theory of Functions (Approxi-mation Theory) (Budapest, 1969), 351–363. Akademiai Kiado, Budapest, 1972.

[12] A. Sperling, Konstanten in den Satzen von Jackson und in den Ungleichun-gen zwischen K-Funktionalen und Stetigkeitsmoduln, Diplomarbeit, UniversitatDuisburg, 1984.

[13] V.V. Zhuk, Functions of the Lip1 class and S. N. Bernstein’s polynomials,(Russian). Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1989, vyp. 1, 25–30,122–123, Vestnik Leningrad Univ. Math. 22 (1989), no. 1, 38-44.

Ana Maria AcuLucian Blaga University of SibiuFaculty of SciencesDepartment of Mathematics and InformaticsStr. Dr. I. Ratiu, No.5-7, 550012 Sibiu, Romaniae-mail: [email protected]

Heiner GonskaUniversity of Duisburg-EssenFaculty of MathematicsForsthausweg 2, 47057 Duisburg, Germanye-mail: [email protected]

Page 27: 2014 Volume 22 No. 1

General Mathematics Vol. 22, No. 1 (2014), 27–41

Modified Bernstein-Durrmeyer operators 1

Tuncer Acar, Ali Aral, Ioan Rasa

Dedicated to the late Academician Professor Dr. Dimitrie D. Stancu

Abstract

In this work, we introduce a new type of Bernstein-Durrmeyer operatorsbased on a function τ (x) which is continuously differentiable ∞− times on[0, 1], such that τ(0) = 0, τ(1) = 1 and τ ′(x) > 0 for x ∈ [0, 1]. We present anasymptotic formula and a quantitative type asymptotic formula for the opera-tors. Later we give comparison with classical Bernstein-Durrmeyer operators.We obtain some direct results for the new operators with the aid of Ditzian-Totik modulus of smoothness. Finally a graphical example is given. All resultsin this work show that our new operators are flexible and sensitive to the rateof convergence to f, depending on our selection of τ (x) .

2010 Mathematics Subject Classification: 41A25, 41A36.

Key words and phrases: Bernstein-Durrmeyer operators, asymptotic formula,local approximation.

1 Introduction

Many well-known operators preserve the linear as well as constant functions. Forexample, Bernstein, Baskakov, Szasz-Mirakyan operators posses these properties,i.e. Ln (ei;x) = ei (x) , where ei (x) = xi (i = 0, 1). To make the convergence fasterKing [18] proposed an approach to modify the classical Bernstein polynomial forf ∈ C [0, 1] by

((Bnf) rn) (x) =n∑

k=0

f

(k

n

)(n

k

)(rn (x))

k (1− rn (x))n−k ,

1Received 19 June, 2014Accepted for publication (in revised form) 29 July, 2014

27

Page 28: 2014 Volume 22 No. 1

28 T. Acar, A. Aral, I. Rasa

where rn is a sequence of continuous functions defined on [0, 1] with 0 ≤ rn (x) ≤ 1for each x ∈ [0, 1] and n ∈ N = 1, 2, ... . King considered a particular case of rn (x)so that the corresponding operators preserve the test functions e0 and e2 of Bohman-Korovkin theorem. Moreover, Gonska et al. [15] constructed sequences of King-typeoperators which are based on a function τ such that τ ∈ C [0, 1] is strictly increasing,τ (0) = 0, τ (1) = 1. These operators are defined by Vn : C [0, 1] → C [0, 1]

V τn f = (Bnf) τn = (Bnf) (Bnτ)

−1 τ.

Inspired by the above ideas, the authors of [6] defined the sequence of linear Bernsteintype operators for f ∈ C[0, 1] by

(1) Bτn (f ;x) =

n∑k=0

(f τ−1

)(kn

)(n

k

)τk (x) (1− τ (x))n−k ,

for any functions τ being continuously differentiable ∞− times on [0, 1], such thatτ(0) = 0, τ(1) = 1 and τ ′(x) > 0 for x ∈ [0, 1]. They investigated its shape preservingand convergence properties as well as its asymptotic behavior and saturation. Thistype of approximation operators generalizes the Korovkin set from 1, e1, e2 to1, τ, τ2

and also presents a better degree of approximation depending on τ.

These advantages of the above construction lead to an interesting area of research,so that generalized Szasz type operators depending on τ and their approximationproperties were recently studied in [2]. In [7], the authors considered the specialcase τ = (e2 + αe1) / (1 + α) and studied some shape preserving approximation andconvergence properties. It was shown that these operators represent a good shapepreserving approximation process making a comparison with Bernstein polynomials.Recently, the authors of [8], using the referred operators, studied some modifiedBernstein-Durrmeyer type operators that reproduce certain test functions.

In the present paper, we deal with Durrmeyer type generalization of (1).Durrmeyer [12] was first who generalized the Bernstein polynomials for integrablefunctions on [0, 1] . Later on, Durrmeyer type operators and further generalizationspresenting better approximation results have been intensively studied. Among theothers, we refer the readers to [9], [8], [10], [17], [16], [21] and the references therein.The operators Bτ

n are meaningful for continuous functions whereas for functionsbelonging to Lebesgue space, the Durrmeyer modifications of them are more useful.In this direction, we consider the following modified Durrmeyer operators:

(2) Dτn (f ;x) = (n+ 1)

n∑k=0

pτn,k (x)

1∫0

(f τ−1

)(t) pn,k (t) dt,

where

pτn,k (x) :=

(n

k

)τk (x) (1− τ (x))n−k

Page 29: 2014 Volume 22 No. 1

Modified Bernstein-Durrmeyer operators 29

and

pn,k (x) :=

(n

k

)xk (1− x)n−k .

If we choose τ (x) = x, we have classical Durrmeyer operators [12] given by

(3) Dn (f ;x) = (n+ 1)

n∑k=0

pn,k (x)

1∫0

f (t) pn,k (t) dt.

The rest of the paper is organized as follows. In the next section, we give someauxiliary results that will be used throughout the paper. Later, we focus on aVoronovskaya type asymptotic formula, as well as its quantitative version, using ageneral construction from which our new operators can be obtained as a special case.We also investigate local approximation properties of Dτ

n in quantitative form usingan appropriate K-functional and Ditzian-Totik moduli of smoothness. Finally wegive a comparison of the new operators with classical Bernstein-Durrmeyer operatorsin terms of a graphical example.

2 Some Lemmas

For our main results, we shall need some auxiliary results. Since they are similarto the corresponding results for the Bernstein-Durrmeyer operators Dn, we give thefollowing lemmas without proofs. Also they can be checked just by taking τ = e1(see [14]).

Lemma 1 We have

Dτne0 = e0, Dτ

nτ =1 + τn

n+ 2, Dτ

nτ2 =

τ2n (n− 1) + 4nτ + 2

(n+ 2) (n+ 3).

Lemma 2 If we define the central moment operator by

µτn,m (x) = Dτn ((τ (t)− τ (x))m ;x)

= (n+ 1)

n∑k=0

pτn,k (x)

1∫0

(t− τ (x))m pn,k (t) dt, m ∈ N,

then we have

µτn,0 (x) = 1, µτn,1 (x) =1− 2τ (x)

n+ 2(4)

µτn,2 (x) =τ (x) (1− τ (x)) (2n− 6) + 2

(n+ 2) (n+ 3).(5)

for all n,m ∈ N.

Page 30: 2014 Volume 22 No. 1

30 T. Acar, A. Aral, I. Rasa

Lemma 3 For all n ∈ N we have

(6) Dτn

(exτ,2;x

)≤ 2

n+ 2δ2n,τ (x) ,

where exτ,i (t) = (τ (t)− τ (x))i and δ2n,τ (x) := φ2τ (x)+

1n+3 , φ

2τ (x) := τ (x) (1− τ (x)) ,

x ∈ [0, 1] .

Lemma 4 For f ∈ C [0, 1] , we have ∥Dτnf∥ ≤ ∥f∥ , where ∥.∥ is the uniform norm

on C [0, 1] .

Proof. By the definition of the operator Dτn and using Lemma 1 we have

|Dτn (f ;x)| ≤ (n+ 1)

n∑k=0

pτn,k (x)

1∫0

∣∣(f τ−1)(t)∣∣ pn,k (t) dt

≤∥∥f τ−1

∥∥Dτn (1;x) = ∥f∥ .

We observe that these operators are positive and linear. Furthermore, in thecase of τ (x) = x, the operators (2) reduce to the classical Bernstein-Durrmeyeroperators. We can also observe that the operators (2) preserve the linear space ofτ−polynomials Pτ

m =τ i : 1 = 1, 2, ...

of degree at most m, that is Dτ

n (Pτm) ⊂ Pτ

m.The uniform convergence of the operators Dτ

n can be obtained by the well knownKorovkin theorem using Lemma 1 and the fact that

e0, τ, τ

2

is an extendedcomplete Tchebychev system on [0, 1] (see [6]). From the result of Shisha and Mond[22] and using Lemma 1 we have

|Dτn (f ;x)− f (x)| ≤

(1 +

µτn,2 (x)

δ2

)ω (f, δ)

for δ > 0, where ω (f, ·) is the classical modulus of smoothness. For τ (x) = x thesimilar result was obtained for the Bernstein-Durrmeyer operator Dn in [14].

We can use this result to compare the rates of convergence of Dτnf and Dn

depending on the selection of the function τ (x) . For example if we choose τ (x) =sin2 πx

2 , we can easily see that

τ (x) (1− τ (x))− x (1− x) ≤ 0

for x ∈ [0, 1] . This inequality tells us that depending on the selection of τ (x) , therate of convergence of Dτ

nf to f is at least so fast as the rate of convergence for theclassical Bernstein-Durrmeyer operator Dn (see Remark 1).

2.1 A General Construction

We first consider a general construction from which we can obtain the operators Dτn

as a special case.

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Modified Bernstein-Durrmeyer operators 31

Suppose that Ln : C [0, 1] → C [0, 1], n ≥ 1, are positive linear operators andLne0 = e0. Let τ ∈ C2 [0, 1] such that τ (0) = 0, τ (1) = 1, τ

′(x) > 0, x ∈ [0, 1] . We

define the operator Kn : C [0, 1] → C [0, 1] by

Kng :=(Ln

(g τ−1

)) τ, n ≥ 1,

for g ∈ C [0, 1] . It is obvious that Kn are linear positive operators and Kne0 = e0.It is easy to verify that

(7) Kmn g :=

(Lmn

(g τ−1

)) τ, m, n ≥ 1,

g ∈ C [0, 1] . Suppose that for f ∈ C [0, 1] there exist Pn : C [0, 1] → C [0, 1] suchthat

(8) limm→∞

Lmn f = Pnf, m ≥ 1,

uniformly on [0, 1] . Then (7) and (8) yield

(9) limm→∞

Kmn g =

(Pn

(g τ−1

)) τ n ≥ 1,

uniformly on [0, 1] .

2.2 Transferring the Asymptotic Formula

For the Bernstein-Durrmeyer operators we have

(10) limn→∞

n (Dn (f, x)− f (x)) = (1− 2x) f ′ (x) + x (1− x) f ′′ (x) .

We use this result to obtain a Voronovskaya type formula for the modified operatorsDτ

n and then the obtained result will be used to compare the operators Dn and Dτn

in the next section.

Theorem 1 Let f ∈ C [0, 1] with f′′(x) finite for x ∈ [0, 1] . If there exist α, β ∈

C [0, 1] such that

(11) limn→∞

n (Ln (f, x)− f (x)) = α (x) f ′′ (x) + β (x) f ′ (x) ,

then we have

(12) limn→∞

n (Kn (g, t)− g (t)) =α (τ (t))

τ ′ (t)2g′′(t)+

(β (τ (t))

τ ′ (t)− α (τ (t)) τ

′′(t)

τ ′ (t)3

)g′(t)

for g ∈ C [0, 1] with g′′(x) finite for x ∈ [0, 1] .

Proof. Since

n (Kn (g, t)− g (t)) = n((Ln

(g τ−1

))(τ (t))−

(g τ−1

)(τ (t))

)

Page 32: 2014 Volume 22 No. 1

32 T. Acar, A. Aral, I. Rasa

we can write from (11)

(13) limn→∞

(Kn (g, t)−g (t))=α (τ (t))(g τ−1

)′′(τ (t))+β (τ (t))

(g τ−1

)′(τ (t)) .

Since (g τ−1

)′=(g′ τ−1

) (τ−1

)′and (

τ−1)′(τ (t)) =

1

τ ′ (t)

we have

(14)(g τ−1

)′(τ (t)) =

g′(t)

τ ′ (t).

Also since (g τ−1

)′′=(g′′ τ−1

)((τ−1

)′)2+(g′ τ−1

) (τ−1

)′′and

d

dt

((τ−1

)′(τ (t))

)=(τ−1

)′′(τ (t)) τ

′(t) = − τ

′′(t)

(τ ′ (t))2

we have

(15)(g τ−1

)′′(τ (t)) =

g′′(t)

(τ ′ (t))2 − g

′(t)

τ′′(t)

(τ ′ (t))3

Using (13), (14) and (15) we obtain the desired result.

2.3 Quantitative type Asymptotic Formula

The estimation of the Peano remainder was given in [19] using the modulus ofcontinuity of n-th derivative f (n) of the function f and the least concave majorantof the modulus. As an application the following quantitative variant of the classicalVoronovskaya theorem was given.

Theorem 2 ([19])Let Ln : C [0, 1] → C [0, 1], n ≥ 1, be positive linear operatorssuch that Lne0 = e0. If f ∈ C2 [0, 1] and x ∈ [0, 1] , then∣∣∣∣Ln (f ;x)− f (x)− f

′(x)Ln ((e1 − x) ;x)− 1

2f

′′(x)Ln

((e1 − x)2 ;x

)∣∣∣∣≤ 1

2Ln

((e1 − x)2 ;x

f ′′,1

3

√√√√√Ln

((e1 − x)4 ;x

)Ln

((e1 − x)2 ;x

) ,

where ω(f

′′, ·)denotes the least concave majorant of ω (f ; ·) given by

(16) ω(f

′′, ε)=

sup0≤x≤ε≤y≤1

(ε−x)ω(f ;y)+(y−ε)ω(f ;x)y−x , if 0 ≤ ε ≤ 1

ω (f, 1) = ω (f ; 1) , if ε > 1.

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Modified Bernstein-Durrmeyer operators 33

2.4 Applications

a) Let Ln = Dn. It is known that (11) is satisfied choosing α (x) = x (1− x) ,β (x) = 1− 2x; see (10). Since Kn = Dτ

n, we get the following theorem.

Theorem 3 If f ∈ C2 [0, 1] , then

limn→∞

n [Dτn (f ;x)− f (x)] =

α (τ (x))

(τ ′ (x))2f ′′ (x) +

(β (τ (x))

τ ′ (x)− α (τ (x)) τ ′′ (x)

(τ ′ (x))3

)f

′(x)

uniformly on [0, 1], with α and β defined above.

b) The iterates of Durrmeyer operators are investigated in [1]. We know from[1, Theorem 3.3, (2)] that

limm→∞

Dmn f =

(∫ 1

0f (x) dx

)e0,

f ∈ C [0, 1] . From (9) it follows for g ∈ C [0, 1] that

limm→∞

(Dτn)

m g =

(∫ 1

0g(τ−1 (t)

)dt

)e0,

uniformly on [0, 1] .

Theorem 4 Let f ∈ C2 [0, 1] . Then the following inequality holds∣∣∣∣[Dτn (f ;x)− f (x)]− f ′ (x)

τ ′ (x)µτn,1 (x)

− 1

τ ′ (x)

(f ′′ (x) τ ′ (x)− f ′ (x) τ ′′ (x)

(τ ′ (x))2

)µτn,2 (x)

∣∣∣∣≤ 1

2µτn,2 (x) ω

((f τ−1

)′′,1

3n−1/2

),

where ω (f, δ) is given in (16).

Proof. If we apply the Theorem 2 with Lnf =(Dn

(f τ−1

)) τ = Dτ

nf , then wehave ∣∣∣∣Dτ

n (f ;x)− f (x)−(f τ−1

)′(τ (x))µτn,1 (x)−

1

2

(f τ−1

)′′(τ (x))µτn,2 (x)

∣∣∣∣≤ 1

2µτn,2 (x) ω

((f τ−1

)′′,1

3n−1/2

).

Corollary 1 If we chose τ(x) = x in Theorem 3, we have the equality (10).

Page 34: 2014 Volume 22 No. 1

34 T. Acar, A. Aral, I. Rasa

3 Local Approximation

To prove the local approximation result let us recall some definitions. For η > 0 andW2 = g ∈ C [0, 1] : g′, g′′ ∈ C [0, 1] , the Peetre’s K-functional [20] is defined by

(17) K (f, η) = infg∈W2

∥f − g∥+ η ∥g∥W2 ,

where

∥f∥W2 = ∥f∥+∥∥f ′∥∥+ ∥∥f ′′∥∥ .

By [4, Proposition 3.4.1], there exists a positive constant C1 > 0 independent of fand η such that

(18) K (f, η) ≤ C1 (ω2 (f,√η) + min 1, η ∥f∥) ,

for all x ∈ [0, 1] and f ∈ C [0, 1], where the second order modulus of continuity of fis defined as

ω2 (f, η) = sup|h|<η

supx,x+2h∈[0,1]

|f (x+ 2h)− f (x+ h) + f (x)| .

The usual modulus of continuity of f ∈ C [0, 1] is defined as

ω (f, η) = sup|h|<η

supx,x+h∈[0,1]

|f (x+ h)− f (x)| .

Throughout the rest of the paper we assume that infx∈[0,1] τ′ (x) ≥ a, a ∈ R+.

Theorem 5 For the operator Dτnf , there exist absolute constants C,Cf > 0 (C is

independent of f and n, Cf is depend only on f) such that

|Dτn (f ;x)− f (x)| ≤ CK

(f,δ2n,τ (x)

(n+ 2)

)+ ω

(f ;

|1− 2τ (x)|(n+ 2) a

).

Proof. We first define an auxiliary operator for f ∈ C [0, 1] by

(19) Dτn (f ;x) = Dτ

n (f ;x) + f (x)−(f τ−1

)(1 + nτ (x)

n+ 2

).

It is clear by Lemma 1 that

(20) Dτn (1;x) = Dτ

n (1;x) = 1,

and

(21) Dτn (τ ;x) = Dτ

n (τ ;x) + τ (x)− 1 + nτ (x)

n+ 2= τ (x) .

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Modified Bernstein-Durrmeyer operators 35

The classical Taylor expansion of the function g ∈ W2 yields for t ∈ [0, 1] that

g (t) =(g τ−1

)(τ (t)) =

(g τ−1

)(τ (x)) +D

(g τ−1

)(τ (x)) (τ (t)− τ (x))

+

τ(t)∫τ(x)

(τ (t)− u)D2(g τ−1

)(u) du.(22)

We can write the last term in the above equality, with change of variable u = τ (y) ,as

τ(t)∫τ(x)

(τ (t)− u)D2(g τ−1

)(u) du

=

t∫x

(τ (t)− τ (y))D2(g τ−1

)τ (y) τ ′ (y) dy.(23)

From (15)

D2(g τ−1

)(τ (y)) =

1

τ ′ (y)

[g′′ (y) τ ′ (y)− g′ (y) τ ′′ (y)

(τ ′ (y))2

],

we have

τ(t)∫τ(x)

(τ (t)− u)D2(g τ−1

)(u) du(24)

=

t∫x

(τ (t)− τ (y))

[g′′ (y) τ ′ (y)− g′ (y) τ ′′ (y)

(τ ′ (y))2

]dy

=

τ(t)∫τ(x)

(τ (t)− u)g′′(τ−1 (u)

)(τ ′ (τ−1 (u)))2

du−τ(t)∫

τ(x)

(τ (t)− u)g′(τ−1 (u)

)τ ′′(τ−1 (u)

)(τ ′ (τ−1 (u)))3

du.

Applying the operator Dτn to both sides of equality (22) and using (21) and (24), we

deduce

Dτn (g;x) = g (x) + Dτ

n

τ(t)∫τ(x)

(τ (t)− u)g′′(τ−1 (u)

)(τ ′ (τ−1 (u)))2

du;x

−Dτ

n

τ(t)∫τ(x)

(τ (t)− u)g′(τ−1 (u)

)τ ′′(τ−1 (u)

)(τ ′ (τ−1 (u)))3

du;x

Page 36: 2014 Volume 22 No. 1

36 T. Acar, A. Aral, I. Rasa

= g (x) +Dτn

τ(t)∫τ(x)

(τ (t)− u)g′′(τ−1 (u)

)(τ ′ (τ−1 (u)))2

du;x

1+nτ(x)n+2∫

τ(x)

(1 + nτ (x)

n+ 2− u

)g′′(τ−1 (u)

)(τ ′ (τ−1 (u)))2

du

+Dτn

τ(t)∫τ(x)

(τ (t)− u)g′(τ−1 (u)

)τ ′′(τ−1 (u)

)(τ ′ (τ−1 (u)))3

du;x

1+nτ(x)n+2∫

τ(x)

(1 + nτ (x)

n+ 2− u

)g′(τ−1 (u)

)τ ′′(τ−1 (u)

)(τ ′ (τ−1 (u)))3

du.

Since τ is strictly increasing on the interval (0, 1) and infx∈[0,1] τ′ (x) ≥ a, a ∈ R+,

we get

∣∣∣Dτn (g;x)− g (x)

∣∣∣ ≤ Dτn

(exτ,2;x

) [∥g′′∥a2

+∥g′∥ ∥τ ′′∥

a3

]+

(1 + nτ (x)

n+ 2− τ (x)

)2 [∥g′′∥a2

+∥g′∥ ∥τ ′′∥

a3

].

We also can write with (6) that

∣∣∣Dτn (g;x)− g (x)

∣∣∣ ≤[∥g′′∥a2

+∥g′∥ ∥τ ′′∥

a3

]2

(n+ 2)δ2n,τ (x)

+

(1 + nτ (x)

n+ 2− τ (x)

)2 [∥g′′∥a2

+∥g′∥ ∥τ ′′∥

a3

]=

[∥g′′∥a2

+∥g′∥ ∥τ ′′∥

a3

]2

(n+ 2)δ2n,τ (x)

+

(1− 2τ (x)

n+ 2

)2 [∥g′′∥a2

+∥g′∥ ∥τ ′′∥

a3

]≤ 3

(n+ 2) a2δ2n,τ (x)

∥∥g′′∥∥+ 3

(n+ 2) a3δ2n,τ (x)

∥∥g′∥∥ ∥∥τ ′′∥∥ .(25)

Furthermore, by Lemma 4, we have

(26)∣∣∣Dτ

n (g;x)∣∣∣ ≤ |Dτ

n (f ;x)|+ |f (x)|+∣∣∣∣(f τ−1

)(1 + nτ (x)

n+ 2

)∣∣∣∣ ≤ 3 ∥f∥ .

Page 37: 2014 Volume 22 No. 1

Modified Bernstein-Durrmeyer operators 37

Hence, for f ∈ C [0, 1] and g ∈ W2, we obtain

|Dτn (f ;x)− f (x)|

=

∣∣∣∣Dτn (f ;x)− f (x) +

(f τ−1

)(1 + nτ (x)

n+ 2

)− f (x)

∣∣∣∣≤

∣∣∣Dτn (f − g;x)

∣∣∣+ ∣∣∣Dτn (g;x)− g (x)

∣∣∣+ |g (x)− f (x)|

+

∣∣∣∣(f τ−1)(1 + nτ (x)

n+ 2

)−(f τ−1

)(τ (x))

∣∣∣∣≤ 4 ∥f − g∥+ 3

(n+ 2) a2δ2n,τ (x)

∥∥g′′∥∥+ 3

(n+ 2) a3δ2n,τ (x)

∥∥g′∥∥ ∥∥τ ′′∥∥+ω

(f τ−1;

∣∣∣∣1− 2τ (x)

n+ 2

∣∣∣∣) ,and if we choose C := max

4, 3

a2, 3∥τ

′′∥a3

then we have

|Dτn (f ;x)− f (x)| ≤ C

∥f − g∥+

δ2n,τ (x) ∥g′′∥n+ 2

+δ2n,τ (x) ∥g′∥

n+ 2+δ2n,τ (x) ∥g∥n+ 2

(f τ−1;

∣∣∣∣1− 2τ (x)

n+ 2

∣∣∣∣) .(27)

On the other hand,

ω(f τ−1; t

)= sup

∣∣f (τ−1 (y))− f

(τ−1 (x)

)∣∣ : 0 ≤ y − x ≤ t

= sup |f (y)− f (x)| : 0 ≤ τ (y)− τ (x) ≤ t .

If 0 ≤ τ(y) − τ(x) ≤ t, then 0 ≤ (y − x) τ ′ (u) ≤ t, for some u ∈ (x, y), i.e.,0 ≤ y − x ≤ t

τ ′(u) ≤ta . Thus we have

ω(f τ−1; t

)≤ sup

|f (y)− f (x)| : 0 ≤ y − x ≤ t

a

= ω

(f ;t

a

).(28)

Using (28) in (27) we have

|Dτn (f ;x)− f (x)| ≤ C

∥f − g∥+

δ2n,τ (x) ∥g′′∥n+ 2

+δ2n,τ (x) ∥g′∥

n+ 2+δ2n,τ (x) ∥g∥n+ 2

(f ;

1

a

∣∣∣∣1− 2τ (x)

n+ 2

∣∣∣∣) .Taking the infimum on the right hand side over all g ∈ W2 we obtain

(29) |Dτn (f ;x)− f (x)| ≤ CK

(f,δ2n,τ (x)

n+ 2

)+ ω

(f ;

|1− 2τ (x)|(n+ 2) a

).

Page 38: 2014 Volume 22 No. 1

38 T. Acar, A. Aral, I. Rasa

Corollary 2 On the other hand, if we use the relation (18) in (29) we get

|Dτn (f ;x)− f (x)| ≤ CC1

[ω2

(f,

δn,τ (x)

(n+ 2)12

)+min

1,δ2n,τ (x)

n+ 2

∥f∥

]

(f ;

|1− 2τ (x)|(n+ 2) a

),

Since

δ2n,τ (x)

n+ 2=τ (x) (1− τ (x)) + 1

n+2

n+ 2≤

1 + 1n+2

n+ 2=

n+ 3

(n+ 2)2< 1

we can write(30)

|Dτn (f ;x)− f (x)| ≤ CC1

[ω2

(f,

δn,τ (x)

(n+ 2)12

)+δ2n,τ (x)

n+ 2∥f∥

]+ω

(f ;

|1− 2τ (x)|(n+ 2) a

).

Remark 1 Let us give some examples for these approximation processes. Letf (x) = x1/4 sin (9x), τ (x) = x2+x

2 . When we choose n = 20, the approximationto the function f by Dn and Dτ

n is shown in the following figure.

Let us take f (x) = ex2, τ (x) = x98. We compute the error of approximation by

using the modulus of continuity for Dn and Dτn at the point x0 = 0.2 in the following

Page 39: 2014 Volume 22 No. 1

Modified Bernstein-Durrmeyer operators 39

table.

n Estimation for the operator Dn Estimation for the operators Dτn

10 1.561785518 1.222692211

102 0.624679640 0.475437534

103 0.216656743 0.164213119

104 0.070701794 0.053564561

105 0.022586191 0.017110834

106 0.007165514 0.005428422

107 0.002268260 0.001718379

108 0.000717519 0.000543575

109 0.000226921 0.000171910

1010 0.000071761 0.000054365

If we compare the results, we can say that Dτn convergence faster than Dn to the

function f at the point x0 = 0, 2.

Acknowledgment. The work was done jointly, while the third author visitedKırıkkale University during May 2014, supported by The Scientific and TechnologicalResearch Council of Turkey.

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Modified Bernstein-Durrmeyer operators 41

Tuncer AcarKırıkkale UniversityFaculty of Science and ArtsDepartment of MathematicsYahsihan, 71450, Kırıkkale, Turkeye-mail: [email protected]

Ali AralKırıkkale UniversityFaculty of Science and ArtsDepartment of MathematicsYahsihan, 71450, Kırıkkale, Turkeye-mail: [email protected]

Ioan RasaTechnical University of Cluj-NapocaCluj-Napoca, Romaniae-mail: [email protected]

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General Mathematics Vol. 22, No. 1 (2014), 43–47

Some Hermite-Hadamard inequalities for convexfunctions on the co-ordinates 1

Aurelia Florea, Eugen Paltanea

Dedicated to the late Academician Professor Dr. Dimitrie D. Stancu

Abstract

The paper deals with the weighted Hermite-Hadamard-type inequalities. Westudy the case of co-ordinated convex functions defined on multi-dimensionalintervals. We present a natural extension for n-dimensional intervals of aninequality due to Fink. In the same framework, we also treat the special caseof convex-concave symmetric functions.

2010 Mathematics Subject Classification: 26D15, 26D99.Key words and phrases: convex function, co-ordinated convex function,

Hermite-Hadamard’s inequality, Fejer’s inequality.

1 Introduction

The first weighted version of the famous Hermite-Hadamard’s inequality, due toFejer [6], is the following:

(1) f

(a+ b

2

)≤∫ ba f(x)p(x) dx∫ b

a p(x) dx≤ f (a) + f (b)

2,

where f : [a, b] → R is a convex function and p : [a, b] → [0,∞) is an integrable

and symmetric about a+b2 function, with

∫ ba p(x) dx > 0. The Hermite-Hadamard’s

inequality is obtained by taking p = 1.On the basis of the Choquet’s theory, a general Hermite-Hadamard-type inequa-

lity can be formulated in the framework of positive Radon measures (see [10]). Fink[7] establishes a relevant weighted version of the Hermite-Hadamard’s inequality

1Received 11 June, 2014Accepted for publication (in revised form) 20 August, 2014

43

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44 A. Florea, E. Paltanea

for real Borel measures, subject to some positivity conditions (see also [8]). Inparticular, the Fink’s result provides the following inequalities

(2) f(m) ≤∫ b

af(x)p(x) dx ≤ b−m

b− af(a) +

m− a

b− af(b),

for a convex function f : [a, b] → R and for an integrable function p : [a, b] → [0,∞),

such that∫ ba p(x) dx = 1 (namely, p is a density function of a random variable X

taking values in [a, b]), where m =∫ ba xp(x) dx is the barycenter of p (or m = E[X]).

Note that the above inequalities are sharp. Also remark that (2) strongly extends(1).

Such kinds of inequalities are treated by a very rich literature. In this paper westudy the multi-dimensional case. More specifically, we refer to the convex functionson the co-ordinates.

Definition 1 Let ∆ := ×ni=1

[a(0)i , a

(1)i

]be a n-dimensional interval of Rn. A func-

tion f : ∆ → R is said to be convex on the co-ordinates (componentwise convex) on∆ if f is convex in each argument when the other arguments are held fixed, that isthe inequality

f (x1, · · · , xi−1, (1− λ)u+ λv, xi+1, · · · , xn)

≤ (1− λ)f (x1, · · · , xi−1, u, xi+1, · · · , xn) + λf (x1, · · · , xi−1, v, xi+1, · · · , xn)

holds for all i ∈ 1, · · · , n, xk ∈[a(0)k , a

(1)k

](k = i), u, v ∈

[a(0)i , a

(1)i

], and λ ∈

[0, 1].

Clearly, any convex function is also convex on the co-ordinates, but the reversestatement is not true. A Hermite-Hadamard-type inequality for convex functions onthe co-ordinates in a bi-dimensional interval has been proved by Dragomir [4]. Inthe same framework, some Fejer-type inequalities for double integrals were recentlyobtained by Alomari and Darus [1] and Latif [9]. Many results on multi-dimensionalHermite-Hadamard inequalities can also be found in [5]. In the spirit of the Fink’sinequalities (2), Cal and Carcamo [2] provide a very nice and instructive probabilisticapproach of weighted Hermite-Hadamard inequalities for convex functions definedon compact convex sets of Rn.

Our purpose is to present some new Hermite-Hadamard inequalities for con-vex functions on the co-ordinates. So, we obtain a natural extension of (2) for n-dimensional intervals. We also discuss a special case of co-ordinated convex-concavefunctions.

2 Main results

The Fink’s result (2) admits the following extension for convex on the co-ordinatesfunctions in n-dimensional intervals.

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Some Hermite-Hadamard inequalities for convex functions ... 45

Theorem 1 Assume a n-dimensional interval ∆ =n∏

i=1

[a(0)i , a

(1)i

]in Rn and

f : ∆ → R a convex on the co-ordinates function. Let pi :[a(0)i , a

(1)i

]→ [0,∞)

be the density function of a random variable Xi, with the mean mi = E [Xi] =∫ a(1)i

a(0)i

xpi(x) dx, for all i ∈ 1, · · · , n. Denote λ(k)i =

(mi − a

(1−k)i

)/(a(k)i − a

(1−k)i

),

for i = 1, · · · , n, k = 0, 1. Then

f (m1, · · · ,mn) ≤∫∆f (x1, · · · , xn)

n∏i=1

pi (xi) dx1 · · · dxn

≤∑

(k1,··· ,kn)∈0,1n

(n∏

i=1

λ(ki)i

)f(a(k1)1 , · · · , a(kn)n

).

Proof. We will prove the result by induction. For n = 1, the conclusion directlyfollows from the relation (2). Suppose that the property is true for a positive in-

teger n. Let ∆1 = ∆ ×[a(0)n+1, a

(1)n+1

]a n + 1-dimensional interval in Rn+1, where

∆ =n∏

i=1

[a(0)i , a

(1)i

]⊂ Rn. Let us consider the density functions pi :

[a(0)i , a

(1)i

]→

[0,∞) with associated means mi, for all i ∈ 1, · · · , n, n+ 1, and define the coeffi-

cients λ(k)i as in the enunciation of the theorem. Assume a convex on the co-ordinates

function f : ∆1 → R. From above assumptions, we have

pn+1 (xn+1) f (m1, · · · ,mn, xn+1)(3)

≤ pn+1 (xn+1)

∫∆f (x1, · · · , xn, xn+1)

n∏i=1

pi (xi) dx1 · · · dxn

≤ pn+1 (xn+1)∑

(k1,··· ,kn)∈0,1n

(n∏

i=1

λ(ki)i

)f(a(k1)1 , · · · , a(kn)n , xn+1

),

for each fixed number xn+1 ∈[a(0)n+1, a

(1)n+1

].

Since f (m1, · · · ,mn, ·) :[a(0)n+1, a

(1)n+1

]→ R is convex, we obtain

f (m1, · · · ,mn,mn+1) ≤∫ a

(1)n+1

a(0)n+1

pn+1 (xn+1) f (m1, · · · ,mn, xn+1) dxn+1,

as a consequence of (2). For similar reasons, we find∫ a(1)n+1

a(0)n+1

pn+1 (xn+1)∑

(k1,··· ,kn)∈0,1nf(a(k1)1 , · · · , a(kn)n , xn+1

) n∏i=1

λ(ki)i dxn+1

=∑

(k1,··· ,kn)∈0,1n

n∏i=1

λ(ki)i

∫ a(1)n+1

a(0)n+1

f(a(k1)1 , · · · , a(kn)n , xn+1

)pn+1 (xn+1) dxn+1

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46 A. Florea, E. Paltanea

≤∑

(k1,··· ,kn)∈0,1n

n∏i=1

λ(ki)i

[a(1)n+1 −mn+1

a(1)n+1 − a

(0)n+1

f(a(k1)1 , · · · , a(kn)n , a

(0)n+1

)

+mn+1 − a

(0)n+1

a(1)n+1 − a

(0)n+1

f(a(k1)1 , · · · , a(kn)n , a

(1)n+1

)]

=∑

(k1,··· ,kn,kn+1)∈0,1n

(n+1∏i=1

λ(ki)i

)f(a(k1)1 , · · · , a(kn)n , a

(kn+1)n+1

).

Now, we integrate (3) on[a(0)n+1, a

(1)n+1

]and apply the Fubini’s theorem for the middle

term. Then, from above last relations, we get the conclusion for the n+1-dimension.This ends the proof by induction of the theorem.

Remark that the result of our theorem can be deduced from Corollary 2 of [2](where the Hermite-Hadamard majorant is not presented in an explicit form), butonly under the more restrictive assumption of the usual convexity.

P. Czinder and Z. Pales [3] proved a Hermite-Hadamard inequality for a particu-lar class of symmetric convex-concave functions. Florea and Niculescu [8]) extendedthis result in the Fink’s meaning. A version of this extension is presented below.

For an interval [a, b] and a fixed point c ∈ [(a + b)/2, b], let us consider adensity function p : [a, b] → [0,∞), with the mean m and the symmetry propertyp(x) = p(2c − x), ∀ x ∈ [2c − b, b]. Assume f : [a, b] → R, such that f(x)+f(2c − x) = 2f(c), ∀ x ∈ [2c − b, b]. In what follows, we will say that the functionf is c−symmetric in x ∈ [2c−b, b]. If f is convex over the interval [a, c] and concaveover the interval [c, b], then the inequalities (2) hold.

Using similar arguments as in the proof of Theorem 1, we obtain the followingn-dimensional extension of Fink’s inequality for convex-concave symmetric functions(the proof is omitted).

Theorem 2 We keep the notations of Theorem 1. Let ci be a fixed point in[(a(0)i + a

(1)i

)/2, a

(1)i

], for i = 1, · · · , n. Suppose that the density function pi on the

interval[a(0)i , a

(1)i

], with the mean mi, has the symmetry property

pi (xi) = pi (2ci − xi) , ∀ xi ∈[2ci − a

(1)i , a

(1)i ,], for all i. Also, assume that, for all

i, the function f (x1, · · · , xi−1, t, xi+1, · · · , xn) is ci-symmetric in t ∈[2ci − a

(1)i , a

(1)i

],

for fixed xj , j ∈ 1, · · · , n \ i. If f (x1, · · · , xi−1, t, xi+1, · · · , xn) is convex in t

over the interval[a(0)i , ci

]and concave in t over the interval

[ci, a

(1)i

](for fixed

xj , j = i), then

f (m1, · · · ,mn) ≤∫∆f (x1, · · · , xn)

n∏i=1

pi (xi) dx1 · · · dxn

≤∑

(k1,··· ,kn)∈0,1n

(n∏

i=1

λ(ki)i

)f(a(k1)1 , · · · , a(kn)n

).

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Some Hermite-Hadamard inequalities for convex functions ... 47

Acknowledgements. This work was partially supported by the Grant number19C/2014, awarded in the internal Grant competition of the University of Craiova.

References

[1] M. Alomari, M. Darus, Fejer inequality for double integrals, Facta Universitatis,vol. 24, 2009, 15-28.

[2] J. Cal, J. Carcamo, Multidimensional Hermite-Hadamard inequalities and theconvex order, J. Math. Anal. Appl. 324, 2006, 248-261.

[3] P. Czinder, Z. Pales, An extension of the Hermite-Hadamard inequality and anapplication for Gini and Stolarsky means, , JIPAM vol. 5, no. 2, 2004, Articleno. 42.

[4] S.S. Dragomir, On Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese J. Math., vol. 5, 2001, 775-788.

[5] S.S. Dragomir, C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequa-lities and Applications, RGMIA Monographs, Victoria University, 2001.

[6] L. Fejer, Uber die Fourierreihen, II, Math. Naturwiss. Anz. Ungar. Akad. Wiss.,vol. 24, 1906, 369-390.

[7] A.M. Fink, A best possible Hadamard Inequality, Math. Inequal. Appl., vol. 1,1998, 223-230.

[8] A. Florea, C.P. Niculescu, A Hermite-Hadamard inequality for convex-concavesymmetric functions, Bull. Math. Soc. Sci. Math. Roumanie, vol. 50(98), no. 2,2007, 149-156.

[9] M.A. Latif, On some Fejer-type inequalities for double integrals, Tamkang J.Math., vol. 43, no. 3, 2012, 423-436.

[10] C.P. Niculescu, L.-E. Persson, Convex Functions and their Applications. A Con-temporary Approach, CMS Books in Mathematics vol. 23, Springer-Verlag, New-York, 2006.

Aurelia Florea Eugen PaltaneaUniversity of Craiova University Transilvania of BrasovDepartment of Applied Mathematics Faculty of Mathematics and InformaticsStr. A. I. Cuza, Nr. 13, Department of Mathematics and InformaticsCraiova, Cod 200585 Str.Iuliu Maniu, Nr. 50, Brasov, Cod 500091e-mail: aurelia [email protected] e-mail: [email protected]

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General Mathematics Vol. 22, No. 1 (2014), 49–58

A Bernstein-Durrmeyer operator which preserves e0and e2 1

Marius Mihai Birou

Dedicated to the late Academician Professor Dr. Dimitrie D. Stancu

Abstract

In this paper we construct a Bernstein-Durrmeyer operator which preservese0 and e2. We give a recurrence relation for the moments of this operator andwe study the convergence. A Voronovskaya formula is presented. Also, we givecomparisons with the classical Durrmeyer operator related to the approximationorder and the approximation error.

2010 Mathematics Subject Classification: 41A36.Key words and phrases: polynomial operator, moments, Voronovskaya formula,

approximation order.

1 Introduction

Let the polynomial operators of Bernstein type Pn : C[0, 1] → Πn, n ≥ 1 given by

(1) Pnf(x) =

n∑k=0

pn,k(x)λn,k(f), x ∈ [0, 1],

where

pn,k(x) =

(n

k

)xk(1− x)n−k, k = 0, ..., n.

Examples of such operators which preserve some test functions ei(x) = xi,i ∈ I ⊂ N (i.e. Pn(ei) = ei) are:· if

λn,k(f) = f

(k

n

), k = 0, ..., n,

1Received 1 July, 2014Accepted for publication (in revised form) 3 September, 2014

49

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50 M.M. Birou

then we get the classical Bernstein operator Bn; it preserves the functions e0 and e1· if

λn,k(f) = f

(√k(k − 1)

n(n− 1)

), k = 0, ..., n,

then we get an operator which preserves the functions e0 and e2 (see I. Gavrea [6])· if

λn,k(f) = f

((k(k − 1) · ... · (k − j + 1)

n(n− 1) · ... · (n− j + 1)

)1/j), k = 0, ..., n,

then we get the Bernstein type operator Bn,0,j considered in [1]; it preserves thefunctions e0 and ej· if

λn,k(f) = (n+ 1)

∫ 1

0pn,k(t)f(t)dt, k = 0, ..., n,

then we get the Durrmeyer operator Mn introduced by J. L. Durrmeyer in [4] andstudied by M. Deriennic (see [3]); it preserves the function e0· if

λn,0(f) = f(0), λn,n(f) = f(1),

λn,k(f) = (n− 1)

∫ 1

0pn−2,k−1(t)f(t)dt, k = 1, ..., n− 1,

then we get the genuiune Bernstein-Durrmeyer operator Un introduced by W. Chenin [2] and T.N.T. Goodman, A. Sharma in [7]; it preserves the functions e0 and e1.

In this article we present a Bernstein-Durrmeyer operator which preserves thetest functions e0 and e2. In Section 2 we construct the operator and we give recu-rrence formulas for the images of the monomials and for the moments of thisoperator. In Section 3 we study the convergence of the operator. We present aVoronovskaya formula. Comparisons with the classical Durrmeyer operator fromthe point of view of the approximation order and the approximation error are givenin Section 4.

2 The construction of the operator and the moments

We look after a Bernstein type approximation operator Pn of the form (1) with

λn,k(f) = c

∫ 1

0xa(1− x)bf(x)dx,

where a, b, c ∈ R, which preserves the test functions e0 and e2. We need that thefollowing conditions to be satisfied (see Z. Finta [5]):

λn,k(e0) = 1, λn,k(e1) ∈

[k(k − 1)

n(n− 1),

√k(k − 1)

n(n− 1)

],

λn,k(e2) =k(k − 1)

n(n− 1), k = 0, ..., n.

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A Bernstein-Durrmeyer operator which preserves e0 and e2 51

We obtain

a = k − 2, b = n− k − 1, c = (n− 2)

(n− 3

k − 2

), k = 2, ..., n− 1, n > 3.

If we takeλn,0(f) = f(0), λn,1(f) = f(0), λn,n(f) = f(1),

then we get the Bernstein-Durrmeyer operator

Dnf(x) = pn,0(x)f (0) + pn,1(x)f (0)

+(n− 2)

n−1∑k=2

pn,k(x)

∫ 1

0pn−3,k−2(t)f(t)dt+ pn,n(x)f (1) , x ∈ [0, 1].

We have

Dne0(x) = 1,(2)

Dne1(x) =nx− 1 + (1− x)n

n− 1,(3)

Dne2(x) = x2.(4)

Theorem 1 The following recurrence relation for the images of the monomialsunder the operator Dn holds

Dnej(x) =nx+ j − 2

n+ j − 2Dnej−1(x) +

x(1− x)

n+ j − 2(Dnej−1)

′(x),

for j ≥ 2.

Proof. We have

Dnej(x) =

n∑k=2

pn,k(x)(k − 1)...(k + j − 2)

(n− 1)...(n+ j − 2).

Using the equalities

(5) x(1− x)p′n,k(x) = (k − nx)pn,k(x), k = 0, ..., n,

we obtain

k + j − 2

n+ j − 2pn,k(x) =

nx+ j − 2

n+ j − 2pn,k(x) +

x(1− x)

n+ j − 2p′n,k(x).

It follows that

Dnej(x) =nx+ j − 2

n+ j − 2

n∑k=2

pn,k(x)(k − 1)...(k + j − 3)

(n− 1)...(n+ j − 3)

+x(1− x)

n+ j − 2

n∑k=2

p′n,k(x)(k − 1)...(k + j − 3)

(n− 1)...(n+ j − 3),

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52 M.M. Birou

and the recurrence relation is provided.The moments of the operator Dn are defined by

Tn,m(x) = Dn((e1 − xe0)m)(x).

Theorem 2 We have the following recurrence relation for the moments of theoperator Dn:

(6) (n+m− 1)Tn,m+1(x) = x(1− x) (Tn,m(x))′

+2mx(1− x)Tn,m−1(x) + (m(1− 2x)− 1 + x)Tn,m(x) + (1− x)n(−x)m, m ≥ 1.

Proof. Let

T 1n,m(x) = pn,0(x)(−x)m + pn,1(x)(−x)m + pn,n(x)(1− x)m

and

T 2n,m(x) = (n− 2)

n−1∑k=2

pn,k(x)

∫ 1

0pn−3,k−2(t)(t− x)mdt.

It follows thatTn,m(x) = T 1

n,m(x) + T 2n,m(x).

We observe that T 1n,m(x) satisfies the recurrence relation (6).

Next, we will show that

(7) (n+m− 1)T 2n,m+1(x)

= x(1− x)(T 2n,m(x)

)′+ 2mx(1− x)T 2

n,m−1(x) + (m(1− 2x)− 1 + x)T 2n,m(x).

We have

(8) (T 2m,n(x))

′ = (n− 2)n−1∑k=2

p′n,k(x)

∫ 1

0pn−3,k−2(t)(t− x)mdt−mT 2

n,m−1(x).

Using the formula (5) we get

(9) x(1− x)n−1∑k=2

p′n,k(x)

∫ 1

0pn−3,k−2(t)(t− x)mdt

=n−1∑k=2

(k − nx)pn,k(x)

∫ 1

0pn−3,k−2(t)(t− x)mdt.

From the equalities

k − nx = k − nt+ n(t− x)

= (k − 2)− (n− 3)t− 3(t− x)− 3x+ 2 + n(t− x)

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A Bernstein-Durrmeyer operator which preserves e0 and e2 53

the last sum from (9) becomes

(10)

n−1∑k=2

pn,k(x)

∫ 1

0t(1− t)p′n−3,k−2(t)(t− x)mdt

+((2− 3x)T 2n,m(x) + (n− 3)T 2

n,m+1(x))/(n− 2).

From (8), (9) and (10) we get

x(1− x)(T 2n,m(x))′ +mx(1− x)T 2

n,m−1(x)− (2− 3x)T 2n,m(x)− (n− 3)T 2

n,m+1(x)

= (n− 2)

n−1∑k=2

pn,k(x)

∫ 1

0t(1− t)p′n−3,k−2(t)(t− x)mdt.

Using the equality

t(1− t) = −(x− t)2 − (1− 2x)(x− t) + x(1− x)

we obtain

(n− 2)n−1∑k=2

pn,k(x)

∫ 1

0p′n−3,k−2(t)× (−(t− x)m+2 + (1− 2x)(t− x)m+1 + x(1− x)(t− x)m)dt

= −(n− 2)n−1∑k=2

pn,k(x)

∫ 1

0pn−3,k−2(t)

× (−(m+ 2)(t− x)m+1 + (m+ 1)(1− 2x)(t− x)m +mx(1− x)(t− x)m−1)dt

= (m+ 2)T 2n,m+1(x)− (m+ 1)(1− 2x)T 2

n,m(x)−mx(1− x)T 2n,m−1(x).

It follows that (7) holds and the proof is completed.

We list the moments up to order four

Tn,0(x) = 1,

Tn,1(x) =x− 1 + (1− x)n

n− 1,

Tn,2(x) = −2x(x− 1 + (1− x)n)

n− 1,

Tn,3(x) =3x2(2(x− 1) + (n+ 1)(1− x)n)

(n− 1)(n+ 1),

Tn,4(x) = −4x2(x(1− x)n(n+ 1)(n+ 2)− 3n(1− x)2 + 9x2 − 12x+ 3)

(n− 1)(n+ 1)(n+ 2).

Using the recurrence formula we can prove

(11) Tn,m(x) = O(n−[

m+12 ]), x ∈ (0, 1].

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54 M.M. Birou

3 The convergence properties

Theorem 3 For all f ∈ C[0, 1] we have

limn→∞

Dnf = f, uniformly on [0, 1].

Proof. The conclusion follows from (2)-(4) by using the well-known Korovkintheorem.

Theorem 4 If f ∈ C[0, 1], then

(12) |Dnf(x)− f(x)| ≤(1 +

2x(1− x− (1− x)n)

(n− 1)δ2

)ω(f, δ),

where x ∈ [0, 1], δ > 0 and ω(f, ·) is the modulus of continuity

ω(f, δ) = sup|f(x+ h)− f(h)| : x, x+ h ∈ [0, 1], 0 ≤ h ≤ δ.

Proof. From the result of Shisha and Mond [9] we have

|Dnf(x)− f(x)| ≤ |f(x)||Dne0(x)− e0(x)|+(Dne0(x) +

1

δ2Dn(e2 − 2xe1 + x2e0)(x)

)ω(f, δ), x ∈ [0, 1].

Using (2)-(4) we get the conclusion.The following theorem give a Voronovskaya formula for the operator Dn.

Theorem 5 If f ∈ C[0, 1] has a second derivative at a point x ∈ (0, 1), then wehave

(13) limn→∞

n (Dn(f)(x)− f(x)) = (x− 1)f ′(x) + x(1− x)f ′′(x).

Proof. From Taylor formula we have

f(t) = f(x) + (t− x)f ′(x) +(t− x)2

2f ′′(x) + (t− x)2µ(t− x)

where µ is a integrable function, bounded on [−x, 1 − x] and with the propertylimu→0 µ(u) = 0. From the linearity of the operator Dn we obtain

Dnf(x) = f(x) +Dn((· − x))(x)f ′(x) +Dn((· − x)2)

2f ′′(x) +Dn((· − x)2µ(· − x))

andn(Dnf(x)− f(x))

= nDn((· − x))(x)f ′(x) +nDn((· − x)2)

2f ′′(x) + nDn((· − x)2µ(· − x)).

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A Bernstein-Durrmeyer operator which preserves e0 and e2 55

Using the formulas for the moments of order one and two of the operator Dn fromthe end of the Section 2, we get

(14) limn→∞

nDn((· − x))(x) = x− 1,

(15) limn→∞

nDn((· − x)2)(x) = 2x(1− x).

It remains to prove that

(16) limn→∞

Rn(x) = 0,

where

Rn(x) = nDn((· − x)2µ(· − x)).

Let M = supu∈[x,1−x] |µ(u)| and ϵ, δ > 0 such that |µ(u)| < ϵ for every u withthe property |u| < δ. We have

|µ(t− x)| < ϵ+M

δ2(t− x)2

for every t ∈ [0, 1]. It follows

|Rn(x)| ≤ ϵnDn((· − x)2)(x) +M

δ2nDn((· − x)4)(x)

≤ ϵ

2(n− 1)+MC4

nδ2,

where C4 is a constant which follows from (11). For n sufficiently large we get

|Rn(x)| ≤ ϵ.

It follows that (16) is true and therefore the limit in (13) exists.

4 Comparisons with the classical Durrmeyer operator

In this section we present comparisons of the operator Dn with the classicalDurrmeyer operator related to the approximation order and approximation error.

First, we compare the approximation orders. For the classical Durrmeyeroperator we have (see [3])

(17) |Mnf(x)− f(x)| ≤(1 +

2 + 2(n− 3)x(1− x)

(n+ 2)(n+ 3)δ2

)ω(f, δ),

where x ∈ [0, 1] and δ > 0, while for the operator Dn we have estimation given byTheorem 4.

Page 56: 2014 Volume 22 No. 1

56 M.M. Birou

In order to get a better approximation order for the operator Dn we need

(18)2x(1− x)

n− 1≤ 2 + 2(n− 3)x(1− x)

(n+ 2)(n+ 3).

It followsx ∈ [0, x1(n)] ∪ [x2(n), 1]

with

x1(n) =1

2−√

3(7 + 26n+ 15n2)

6(1 + 3n), x2(n) =

1

2+

√3(7 + 26n+ 15n2)

6(1 + 3n).

We have

limn→∞

x1(n) =1

2−

√5

6= 0.1273...

limn→∞

x2(n) =1

2+

√5

6= 0.8726...

Next, we compare the approximation error.

Theorem 6 If f is decreasing and convex on [0, 1], then there exists n1 ∈ N suchthat for n ≥ n1 we have

(19) f(x) ≤ Dnf(x) ≤Mnf(x), x ∈ [2/3, 1].

Proof. As the operator Dn preserves the test functions e0 and e2, from [10, Th. 2]we have

(20) f(x) ≤ Dnf(x), x ∈ [0, 1]

for every f which is convex with respect to the function e2. For more about gene-ralized convexity see [8].

We define the function

(21) g : [0, 1] → R, g(x) = f(√x).

The function f is convex with respect to the function e2 if and only if the functiong is convex in the classical sense. We have

g′′(x) =1

4x√x

(√xf ′′(

√x)− f ′(

√x)), x ∈ (0, 1].

It follows that if f is decreasing and convex then g is convex and the inequality (20)holds (we have equality in 0 due the interpolation property of the operator Dn atthe endpoints).

For the second inequality we compare the Voronovskaya formula for the classicalDurrmeyer operator (see [3])

limn→∞

n (Mnf(x)− f(x)) = (1− 2x)f ′(x) + x(1− x)f ′′(x), x ∈ [0, 1],

with the formula (13).

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A Bernstein-Durrmeyer operator which preserves e0 and e2 57

Theorem 7 Let f ∈ C3[0, 1] be a function satisfying one of the conditionsa)f is increasing on [0, 1] and f ′′′(x) ≥ 0, x ∈ [0, 1] and f ′(0) = 0,b)f is increasing on [0, 1] and f ′′′(x) ≤ 0, x ∈ [0, 1] and f ′′(1)− f ′(1) ≥ 0,then there exists n2 ∈ N such that for n ≥ n2 we have

(22) f(x) ≤ Dnf(x) ≤Mnf(x), x ∈ [0, 2/3].

Proof. We follow the same steps as in the proof of the previous theorem.For the first inequality we study the convexity of the function g given by (21). We

define the function h : [0, 1] → R, h(x) = xf ′′(x) − f ′(x). We have h′(x) = xf ′′′(x),x ∈ [0, 1].

It follows that if one of the conditions a) and b) are satisfied then g is convex.Thus, the first inequality holds.

For the second inequality we compare the Voronovskaya formulas for these twooperators.

References

[1] J. M. Aldaz, O. Kounchev, H. Render, Shape preserving properties of generalizedBernstein operators on extended Chebyshev spaces, Numer. Math., vol. 114,2009, 1-25.

[2] W. Chen, On the modified Bernstein-Durrmeyer operators, Report of the FifthChinese Conference on Approximation Theory, Zhen Zhou, China 1987.

[3] M. M. Derrienic, Sur l’approximation des fonctions integrables sur [0, 1] par despolynomes de Bernstein modifies, J. Approx. Theory, vol. 31, 1981, 325-343.

[4] J. L. Durrmeyer, Une formule d’inversion de la transformee de Laplace: appli-cations a la theorie des moments, These de 3e cycle, Faculte des Sciences del’Universite de Paris, 1967.

[5] Z. Finta, Estimates for Bernstein type operators, Math. Inequal. Appl., vol. 15,no. 1, 2012, 127-135.

[6] I. Gavrea, On Bernstein-Stancu type operators, Stud. Univ. Babes Bolyai, vol.52, no. 4, 2007, 81-88.

[7] T. N. T. Goodman, A. Sharma, A modified Bernstein-Schoenberg operator, in:Proc. of the Conference on Constructive Theory of Functions, Varna 1987 (ed.by Sendov et al.), Sofia: Publ. House Bulg. Acad. of Sci. 1988, 166-173.

[8] S. Karlin, W. Studden, Tchebycheff Systems: with Applications in Analysis andStatistics, Interscience, New York, 1966.

[9] B. Mond, Note: On the degree of approximation by linear positive operators, J.Approx. Theory, vol. 18, 1976, 304-306.

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58 M.M. Birou

[10] Z. Ziegler, Linear approximation and generalized convexity, J. Approx. Theory,vol. 1, 1968, 420-433.

Marius Mihai BirouTechnical University of Cluj NapocaFaculty of Automation and Computer ScienceMathematicsStr. Memorandumului, No. 28e-mail: [email protected]

Page 59: 2014 Volume 22 No. 1

General Mathematics Vol. 22, No. 1 (2014), 59–67

A de Casteljau type algorithm in matrix form 1

Daniela Inoan, Ioan Rasa

Dedicated to the late Academician Professor Dr. Dimitrie D. Stancu

Abstract

We describe a de Casteljau type algorithm in matrix form for some linearoperators that appear in Approximation Theory. Some monotonicity preservingproperties of the operators are proved by using this algorithm.

2010 Mathematics Subject Classification: 41A36, 65F30.

Key words and phrases: linear operators, de Casteljau algorithm, monotonicitypreserving properties.

1 Introduction

For x ∈ [0, 1] and k ∈ N consider the matrix

Ak(x) :=

1− x x 0 . . . 0 0 00 1− x x . . . 0 0 00 0 1− x . . . 0 0 0. . . . . . . . . . . . . . . . . .0 0 0 . . . 1− x x 00 0 0 . . . 0 1− x x

∈ Mk,k+1.

1Received 26 June, 2014Accepted for publication (in revised form) 1 September, 2014

59

Page 60: 2014 Volume 22 No. 1

60 D. Inoan, I. Rasa

Let f ∈ C[0, 1], n ∈ N and Fn :=(f(0), f

(1n

), . . . , f

(n−1n

), f(1)

)t. Denote

Bnf(x) := A1(x)A2(x) . . . An(x)Fn. Then

A1(x) =(1− x, x

)A1(x)A2(x) =

((1− x)2, 2x(1− x), x2

). . . . . . . . .

A1(x)A2(x) . . . An(x) =

((1− x)n,

(n

1

)x(1− x)n−1, . . .

. . . ,

(n

k

)xk(1− x)n−k, . . . , xn

).

It follows that Bnf(x), defined above, is the classical Bernstein polynomial asso-ciated with the function f .

On the other hand,

An(x)Fn, An−1(x)An(x)Fn, . . . , A1(x)A2(x) . . . An(x)Fn

represent the successive steps in the classical de Casteljau algorithm for computingthe value Bnf(x).

These remarks can be found in [2], where they are applied also to the Bernsteinoperators of second kind.

In this paper we generalize the above approach and use the de Casteljau algo-rithm in matrix form to get monotonicity preserving properties for other operatorsappearing in Approximation Theory.

2 Monotonicity preserving properties

For each n ∈ N, we denote by Vn the set of vectors in Rn having the components inincreasing order:

Vn = v = (v1, v2, . . . , vn)t | vj ∈ R, v1 ≤ v2 ≤ · · · ≤ vn

and by V +n the set of vectors in Vn having all the components non-negative

V +n = v = (v1, v2, . . . , vn)

t | vj ∈ R, 0 ≤ v1 ≤ v2 ≤ · · · ≤ vn.

If u = (u1, u2, . . . , un) and v = (v1, v2, . . . , vn) are two vectors in Rn by u ≤ v wemean that uj ≤ vj for all j ∈ 1, . . . n.

For n ∈ N, k ∈ 1, . . . , n and x ∈ [0, 1] let Ak(x) be a matrix with k rows andk+1 columns, with real elements akij(x). If a

kij : [0, 1] → R are continuous functions,

we can define an operator Ln : C[0, 1] → C[0, 1] by

Ln(f)(x) = A1(x)A2(x) . . . An(x)(f(0), f

( 1n

), . . . , f

(n− 1

n

), f(1)

)t,

for every f ∈ C[0, 1], x ∈ [0, 1].

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A de Casteljau type algorithm in matrix form 61

Theorem 1 Suppose that for every k ∈ 1, . . . , n the following conditions aresatisfied:

(H1) If v ∈ Vk+1, then Ak(x)v ∈ Vk, for every x ∈ [0, 1].

(H2) If x, y ∈ [0, 1] such that x < y and u, v ∈ Vk+1 with u ≤ v, thenAk(x)u ≤ Ak(y)v.

Then, if f ∈ C[0, 1] is an increasing function, Ln(f) is increasing too.

Proof. Let f : [0, 1] → R be an increasing function. Let x, y ∈ [0, 1] such thatx < y. We want to prove that Ln(f)(x) ≤ Ln(f)(y), i.e.,

(1)A1(x)A2(x) . . . An(x)

(f(0), f

( 1n

), . . . , f(1)

)t≤ A1(y)A2(y) . . . An(y)

(f(0), f

( 1n

), . . . , f(1)

)t.

Since f is increasing we have that(f(0), f

(1n

), . . . , f(1)

)t∈ Vn+1 and by (H1)

un(x) := An(x)(f(0), f

(1n

), . . . , f(1)

)t∈ Vn and

un(y) := An(y)(f(0), f

(1n

), . . . , f(1)

)t∈ Vn.

Also by (H2) we get un(x) ≤ un(y).

Continuing in the same way it follows that un−1(x) := An−1(x)un(x) ∈ Vn−1,un−1(y) := An−1(y)un(x) ∈ Vn−1 and un−1(x) ≤ un−1(y).

Step by step we get to prove (1).

A similar result can be proved for increasing functions that take only nonnegativevalues:

Theorem 2 Suppose that for every k ∈ 1, . . . , n and every x ∈ [0, 1] the matrixAk(x) has only non-negative elements and the following conditions are satisfied:

(H1’) If v ∈ V +k+1, then Ak(x)v ∈ V +

k , for every x ∈ [0, 1]

(H2’) If x, y ∈ [0, 1] such that x < y and u, v ∈ V +k+1 with u ≤ v, then

Ak(x)u ≤ Ak(y)v.

Let f ∈ C[0, 1] be an increasing function such that f(x) ≥ 0 for every x ∈ [0, 1].Then Ln(f) is increasing and Ln(f)(x) ≥ 0 for every x ∈ [0, 1].

Next we give some characterizations for matrices that satisfy hypotheses (H1) and(H1’).

Theorem 3 Let A = (aij)1≤i≤k,1≤j≤k+1 ∈ Mk,k+1 be a matrix such that aij ≥ 0for all i ∈ 1, . . . , k, j ∈ 1, . . . , k + 1. The following assertions are equivalent:

(i) Av ∈ V +k for every v ∈ V +

k+1.

(ii)( k+1∑

j=l

a1j ,

k+1∑j=l

a2j , . . . ,

k+1∑j=l

akj

)t∈ V +

k , for every l ∈ 1, . . . , k + 1.

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62 D. Inoan, I. Rasa

Proof. (i) ⇒ (ii) Let l ∈ 1, . . . , k + 1. The vector (0, . . . , 0, 1, . . . , 1)t (first l − 1positions equal to zero) belongs to V +

k+1, so according to (i),

A(0, . . . , 0, 1, . . . , 1)t =( k+1∑

j=l

a1j ,

k+1∑j=l

a2j , . . . ,

k+1∑j=l

akj

)t∈ V +

k .

(ii) ⇒ (i) Let v = (x1, . . . , xk+1)t be an arbitrary vector from V +

k+1, i.e., 0 ≤ x1 ≤x2 ≤ · · · ≤ xk+1. Then we can write v in the form

v = x1(1, 1, . . . , 1)t + (x2 − x1)(0, 1, . . . , 1)

t + · · ·+ (xk+1 − xk)(0, 0, . . . , 1)t

and then

Av = x1

k+1∑j=1

a1j , . . . ,

k+1∑j=1

akj

t

+ (x2 − x1)

k+1∑j=2

a1j , . . . ,

k+1∑j=2

akj

t

+ · · ·+ (xk+1 − xk)

k+1∑j=k+1

a1j , . . . ,

k+1∑j=k+1

akj

t

From (ii) and the fact that xl+1−xl ≥ 0 for all l ∈ 1, . . . , k it follows that Av ∈ V +k .

Theorem 4 Let A = (aij)1≤i≤k,1≤j≤k+1 ∈ Mk,k+1 be a matrix such that aij ≥ 0for all i ∈ 1, . . . , k, j ∈ 1, . . . , k + 1. The following assertions are equivalent:

(i) Av ∈ Vk for every v ∈ Vk+1.

(ii)( k+1∑

j=l

a1j ,k+1∑j=l

a2j , . . . ,k+1∑j=l

akj

)t∈ V +

k , for every l ∈ 1, . . . , k + 1 and

k+1∑j=1

a1j =

k+1∑j=1

a2j = · · · =k+1∑j=1

akj = a.

Proof. (i) ⇒ (ii) The first part follows as in the previous result. To prove thesecond part, we consider the vector v = (−1,−1, . . . ,−1)t ∈ Vk+1. From (i) we

have Av ∈ Vk, that is −k+1∑j=1

a1j ≤ −k+1∑j=1

a2j ≤ · · · ≤ −k+1∑j=1

akj . Since we have also

k+1∑j=1

a1j ≤k+1∑j=1

a2j ≤ · · · ≤k+1∑j=1

akj , the equality of the sums follows.

(ii) ⇒ (i) Let v ∈ Vk+1. Then there exists m ≥ 0 such that v+m(1, 1, . . . , 1)t =(x′1, . . . , x

′k+1)

t ∈ V +k+1. This vector can be written as

v +m(1, 1, . . . , 1)t = x′1(1, 1, . . . , 1)t + (x′2 − x′1)(0, 1, . . . , 1)

t

+ · · ·+ (x′k+1 − x′k)(0, 0, . . . , 1)t,

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A de Casteljau type algorithm in matrix form 63

so v = m(−1,−1, . . . ,−1)t + x′1(1, 1, . . . , 1)t + (x′2 − x′1)(0, 1, . . . , 1)

t + . . .+(x′k+1 − x′k)(0, 0, . . . , 1)

t. Then the product between the matrix A and thevector v is

Av = m(−a,−a, . . . ,−a)t + x′1(a, a, . . . , a)t

+ (x′2 − x′1)( k+1∑

j=2

a1j ,

k+1∑j=2

a2j , . . . ,

k+1∑j=2

akj

)t+ . . .

+ (x′k+1 − x′k)(a1,k+1, a2,k+1, . . . , ak,k+1)t

and it belongs to Vk.

3 Applications

I. Let (λn)n≥1, (ρn)n≥1 be two sequences of positive numbers. For x ∈ [0, 1], letA1(x) = (λ1(1− x) ρ1x) and for k ∈ 2, . . . , n

Ak(x) =

λkλk−1

(1− x) x 0 . . . 0 0

0 1− x x . . . 0 00 0 1− x . . . 0 0. . . . . . . . . . . . . . . . . .0 0 0 . . . x 00 0 0 . . . 1− x ρk

ρk−1x

∈ Mk,k+1

It is easy to see that, for any function f ∈ C[0, 1],

A1(x)A2(x) . . . An(x)(f(0), f

( 1n

), . . . , f(1)

)t=

n∑k=0

αn,kxk(1− x)n−kf

(kn

)=:Mn(f)(x),

where αn,0 = λn, αn,n = ρn and αn+1,k = αn,k + αn,k+1 for k ∈ 1, . . . , n.We obtained in this way the operator Mn(f) introduced by Campiti and Meta-

fune in [1] as a generalization of the classical Bernstein operators. Using the resultsof Section 2 we can prove:

Theorem 5 Let (λn)n≥1 be a decreasing sequence of positive numbers and (ρn)n≥1

an increasing sequence of positive numbers, such that λ1 ≤ ρ1. If f : [0, 1] → R isincreasing and f(x) ≥ 0 for each x ∈ [0, 1], then Mn(f) is an increasing functionand Mn(f)(x) ≥ 0 for all x ∈ [0, 1].

Proof. First we check assertion (ii) from Theorem 3 for every Ak, k ∈ 1, . . . , n.

Page 64: 2014 Volume 22 No. 1

64 D. Inoan, I. Rasa

We have( k+1∑j=1

a1j , . . . ,k+1∑j=1

akj

)=( λkλk−1

(1− x) + x, 1, . . . , 1, 1− x+ρkρk−1

x),

( k+1∑j=2

a1j , . . . ,

k+1∑j=2

akj

)=(x, 1, . . . , 1, 1− x+

ρkρk−1

x), . . . . . . ,

( k+1∑j=k

a1j , . . . ,k+1∑j=k

akj

)=(0, 0, . . . , x, 1− x+

ρkρk−1

x),

( k+1∑j=k+1

a1j , . . . ,k+1∑

j=k+1

akj

)=(0, 0, . . . , 0,

ρkρk−1

x)

These vectors belong to V +k if and only if λk ≤ λk−1 and ρk−1 ≤ ρk. Hypothesis

(H1’) follows now directly.We check next hypothesis (H2’) of Theorem 2 for each k ∈ 1, . . . , n. Let x < y

be arbitrary fixed, u, v ∈ V +k+1 with u ≤ v.

For k = 1, let u = (u1, u2)t. A1(x)u ≤ A1(y)u is equivalent to λ1(1 − x)u1 +

ρ1xu2 ≤ λ1(1− y)u1 + ρ1yu2, that is (y − x)(λ1u1 − ρ1u2) ≤ 0. This is true for any0 ≤ u1 ≤ u2 if and only if λ1 ≤ ρ1. Obviously, if u ≤ v then A1(y)u ≤ A1(y)v, soalso A1(x)u ≤ A1(y)v.

For k ∈ 2, . . . , n, Ak(x)u ≤ Ak(y)u means that three types of inequalities haveto be satisfied:

1)λkλk−1

(1 − x)u1 + xu2 ≤ λkλk−1

(1 − y)u1 + yu2, which is equivalent to( λkλk−1

u1 − u2

)(y − x) ≤ 0, insured by the conditions λk ≤ λk−1 and 0 ≤ u1 ≤ u2;

2) (1− x)uj + xuj+1 ≤ (1− y)uj + yuj+1, equivalent to (uj − uj+1)(y − x) ≤ 0which is obviously true;

3) (1 − x)uk +ρkρk−1

xuk+1 ≤ (1 − y)uk +ρkρk−1

yuk+1, which is equivalent to(uk−

ρkρk−1

uk+1

)(y−x) ≤ 0, insured by the conditions ρk ≥ ρk−1 and 0 ≤ uk ≤ uk+1.

Finally, if u ≤ v it follows also that Ak(x)u ≤ Ak(y)v.

Remark 1 The conditions of Theorem 5 are not sufficient to insure the transforma-tion of any increasing function into an increasing function by the Campiti-Metafuneoperator. For instance, if we choose n = 2, λk = 1 and ρk = 4 for all k ∈ 1, 2 theconditions of Theorem 5 are satisfied and we obtain the operator of order two

M2(f)(x) = (1− x)2f(0) + 5x(1− x)f(12

)+ 4x2f(1).

The function f : [0, 1] → R, f(x) = 2x− 2 is increasing but has also negative valuesin [0, 1]. It is transformed into M2(f)(x) = 3x2 − x− 2, which is not an increasingfunction on [0, 1].

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A de Casteljau type algorithm in matrix form 65

Concerning monotonicity preserving for functions with arbitrary values, we have

Theorem 6 The Campiti-Metafune operator Mn transforms any increasing func-tion f : [0, 1] → R into an increasing function if and only if there exists α ≥ 0 suchthat

Mn(f)(x) =n∑

k=0

α

(n

k

)xk(1− x)n−kf

(kn

).

Proof. The constant functions ±1 are increasing, so that α := Mn(1) and −α :=Mn(−1) are increasing functions. It follows that α is a constant function. Then

n∑k=0

αn,kxk(1− x)n−k =Mn(1)(x) = α =

n∑k=0

α

(n

k

)xk(1− x)n−k,

for x ∈ [0, 1]. From the linear independence of the functions xk(1 − x)n−k : k =0, . . . , n we get αn,k = α

(nk

), for k = 0, . . . , n.

II. Another operator that can be described using matrices is an operator ofStancu type.

Let γ ≥ 0 be fixed. For x ∈ [0, 1], k ∈ 1, . . . , n let

Ak(x) =1

1 + (k − 1)γ

b11 b12 0 0 . . . 0 00 b22 b23 0 . . . 0 00 0 b33 b34 . . . 0 0. . . . . . . . . . . . . . . . . . . . .0 0 0 0 . . . bkk bk,k+1

with bjj = 1− x+ (k − j)γ and bj,j+1 = x+ (j − 1)γ.

It is easy to see that, for any function f ∈ C[0, 1],

A1(x) . . . An(x)

(f(0), f

(1

n

), . . . , f(1)

)t

=n∑

k=0

un,k(x)f

(k

n

)=: Sn(f)(x),

where

un,k(x) =

(n

k

)x(x+ γ) . . . (x+(k − 1)γ)(1− x)(1− x+γ) . . . (1− x+ (n− k − 1)γ)

(1 + γ)(1 + 2γ) . . . (1 + (n− 1)γ).

For γ = 0 the classical Bernstein operator is obtained.

Theorem 7 Let γ ≥ 0 be fixed. If the function f : [0, 1] → R is increasing, thenalso Sn(f) : [0, 1] → R is increasing.

Proof. To check hypothesis (H1) of theorem 1 we will use Theorem 4. Letk ∈ 1, . . . , n.

Page 66: 2014 Volume 22 No. 1

66 D. Inoan, I. Rasa

We have

k+1∑j=1

a1j =

k+1∑j=1

a2j = · · · =k+1∑j=1

akj = 1,

( k+1∑j=2

a1j , . . . ,

k+1∑j=2

akj

)=( x

1 + (k − 1)γ, 1, . . . , 1, 1

), . . . . . . ,

( k+1∑j=l

a1j , . . . ,

k+1∑j=l

akj

)=(0, 0, . . . ,

x+ (l − 2)γ

1 + (k − 1)γ, . . . , 1, 1

), . . . . . . ,

( k+1∑j=k+1

a1j , . . . ,

k+1∑j=k+1

akj

)=(0, 0, . . . , 0,

x+ (k − 1)γ

1− (k − 1)γ

),

which belong to the set V +k .

We verify now hypothesis (H2) of Theorem 2, for each k ∈ 1, . . . , n. Letx, y ∈ [0, 1], x < y, u, v ∈ Vk+1 with u ≤ v. The inequality Ak(x)u ≤ Ak(y)u isequivalent to

1− x(k − j)γ

1 + (k − 1)γuj +

x+ (j − 1)γ

1 + (k − 1)γuj+1 ≤

1− y(k − j)γ

1 + (k − 1)γuj +

y + (j − 1)γ

1 + (k − 1)γuj+1,

for all j ∈ 1, . . . , k. This means (y− x)(uj − uj+1 ≤ 0), obviously true. For u ≤ vit is clear that Ak(y)u ≤ Ak(y)v holds. So we have finally Ak(x)u ≤ Ak(y)v asrequested in Theorem 1.

III. Let τ be a strictly increasing function in C[0, 1] and for each k ∈ 1, . . . , ndefine the matrix

Ak(x) =

1− τ(x) τ(x) 0 . . . 0 0

0 1− τ(x) τ(x) . . . 0 00 0 1− τ(x) . . . 0 0. . . . . . . . . . . . . . .0 0 0 . . . 1− τ(x) τ(x)

Using these matrices, the following operator Ln : C[0, 1] → C[0, 1] can be defined

Ln(f)(x) := A1(x) . . . An(x)(f(τ−1(0)), f

(τ−1

( 1n

)). . . f(τ−1(1))

)t=

n∑k=0

(n

k

)τk(x)(1− τ(x))n−kf

(τ−1

(kn

))for any f ∈ C[0, 1]. This is another generalization of the classical Bernstein operator.

Theorem 8 Let τ : [0, 1] → [0, 1] be a strictly increasing function. If the functionf : [0, 1] → R is increasing, then also Ln(f) : [0, 1] → R is increasing.

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A de Casteljau type algorithm in matrix form 67

Proof. If f is increasing, then so is g := f τ−1 and we apply Theorem 1 for thefunction g.

We havek+1∑j=1

a1j =k+1∑j=1

a2j = · · · =k+1∑j=1

akj = 1,

( k+1∑j=2

a1j , . . . ,

k+1∑j=2

akj

)=(τ(x), 1, . . . , 1, 1

)∈ V +

k , . . . . . . ,

( k+1∑j=k+1

a1j , . . . ,k+1∑

j=k+1

akj

)=(0, 0, . . . , 0, τ(x)

)∈ V +

k ,

so by Theorem 4, we have (H1) fulfiled.To check (H2), let x, y ∈ [0, 1], x < y and u, v ∈ Vk+1, u ≤ v. The inequality

Ak(x)u ≤ Ak(y)u is equivalent to

(1− τ(x))uj + τ(x)uj+1 ≤ (1− τ(y))uj + τ(y)uj+1,

i.e., (τ(y) − τ(x))(uj − uj+1) ≤ 0 which is obviously true. Finally, since Ak(y)u ≤Ak(y)v it follows that Ak(x)u ≤ Ak(y)v.

Remark 2 The operators discussed in this section are of King type. Detailsconcerning them, as well as the Campiti-Metafune operators, can be found in [3]and the references therein.

References

[1] M. Campiti, G. Metafune, Approximation properties of recursively definedBernstein-type operators, J. Approx. Theory vol. 87, 1996, 243–269.

[2] D. Inoan, I. Rasa, A recursive algorithm for Bernstein operators of second kind,Numer. Algor. vol. 64 (4), 2013, 699–706.

[3] I. Rasa, Approximation processes and asymptotic relations, Carpathian J. Math(to appear).

Daniela Inoan, Ioan RasaTechnical University of Cluj-NapocaFaculty of Automation and Computer ScienceDepartment of MathematicsStr. Memorandumului nr. 28 Cluj-Napoca, Romaniae-mail: [email protected], [email protected]

Page 68: 2014 Volume 22 No. 1
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General Mathematics Vol. 22, No. 1 (2014), 69–73

Two reverse inequalities of Bullen’s inequality 1

Nicusor Minculete, Petrica Dicu, Augusta Ratiu

Dedicated to the late Academician Professor Dr. Dimitrie D. Stancu

Abstract

In this paper we provide two reverse inequalities of Bullen′s inequality andseveral applications concerning the arithmetic mean and the logarithmic mean.

2010 Mathematics Subject Classification: 26A51, 26D15.

Key words and phrases: Bullen′s inequality, convex function.

1 Introduction

Given a convex function f : [a, b] −→ R, one has Bullen′s inequality (see e.g. [2,3])

(1)2

b− a

∫ b

af(x)dx ≤ f(a) + f(b)

2+ f

(a+ b

2

).

For certain constraints of f , in [1] Dragomir and Pearce found an improvementof Bullen′s inequalit given by the following:

Theorem 1 Let f : [a, b] −→ R be a twice differentiable function for wich thereexist real constants m and M such that:

m ≤ f ′′(x) ≤M, for all x ∈ [a, b].

Then

(2) m(b− a)2

24≤ f(a) + f(b)

2+ f

(a+ b

2

)− 2

b− a

∫ b

af(x)dx ≤M

(b− a)2

24.

1Received 25 June, 2014Accepted for publication (in revised form) 15 August, 2014

69

Page 70: 2014 Volume 22 No. 1

70 N. Minculete, P. Dicu, A. Ratiu

2 Main results

Lemma 1 Whenever f : [a, b] −→ R is a twice differentiable function, we have thefollowing equality:(3)f(a) + f(b)

2+ f(

a+ b

2)− 2

b− a

∫ b

af(x)dx =

1

b− a

∫ b

a

(x− a+ b

2

)q(x)f ′′(x)dx

where

q(x) :=

a− x , x ∈

[a,a+ b

2

)b− x , x ∈

[a+ b

2, b

]Proof. It is easy to see that

1

b− a

∫ b

a

(x− a+ b

2

)q(x)f ′′(x)dx =

1

b− a

∫ a+b2

a

(x− a+ b

2

)(a− x)f ′′(x)dx

+1

b− a

∫ b

a+b2

(x− a+ b

2

)(b− x)f ′′(x)dx

=1

b− a

[(x− a+ b

2

)(a− x)f ′(x)

∣∣∣∣a+b2

a

−∫ a+b

2

af ′(x)

(− 2x+

3a+ b

2

)dx

+

(x− a+ b

2

)(b− x)f ′(x)

∣∣∣∣ba+b2

−∫ b

a+b2

f ′(x)

(− 2x+

a+ 3b

2

)dx

]

=1

b− a

[ ∫ a+b2

af ′(x)

(2x− 3a+ b

2

)dx+

∫ b

a+b2

f ′(x)

(2x− a+ 3b

2

)dx

]

=1

b− a

[f(x)

(2x− 3a+ b

2

)∣∣∣∣a+b2

a

− 2

∫ a+b2

af(x)dx+ f(x)

(2x− a+ 3b

2

)∣∣∣∣ba+b2

−2

∫ b

a+b2

f(x)dx

]=f(a) + f(b)

2+ f

(a+ b

2

)− 2

b− a

∫ b

af(x)dx.

Remark 1 1. a) Clearly for x ∈ [a, b], one has(x− a+ b

2

)q(x) ≥ 0.

By some elementary computations one obtains:∫ b

a

(x− a+ b

2

)q(x) =

(b− a)3

24.

Therefore, for every x ∈ [a, b], we can write

m

(x− a+ b

2

)q(x) ≤

(x− a+ b

2

)q(x)f ′′(x) ≤M

(x− a+ b

2

)q(x).

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Two reverse inequalities of Bullen’s inequality 71

Integrating from a to b, multiplying by1

b− aand using relation (3), we obtain

the inequalities from (2).b) Inequalities (2) can also be obtained from inequality (1) by applying the Bullen′s

inequality for the convex function f(x)−mx2

2and M

x2

2− f(x).

In the following we give a reverse inequality of Bullen′s inequality.

Theorem 2 Let f : [a, b] −→ R be a twice differentiable and convex function. Thenthe following inequality holds

(4)f(a) + f(b)

2+ f(

a+ b

2)− 2

b− a

∫ b

af(x)dx ≤ (b− a)[f ′(b)− f ′(a)]

16

Proof. Since f is a convex function, it follows that f ′′(x) ≥ 0, for every x ∈ [a, b].Because

0 ≤(x− a+ b

2

)q(x) ≤ (b− a)2

16

for any x ∈ [a, b] we deduce the inequality(x− a+ b

2

)q(x)f ′′(x) ≤ (b− a)2

16f ′′(x).

Therefore, by integrating, the last inequality from a to b, we obtain:∫ b

a

(x− a+ b

2

)q(x)f ′′(x)dx ≤ (b− a)2

16[f ′(b)− f ′(a)].

Using equality (3) in the previous inequality we find the inequality from statement.

Theorem 3 Let f : [a, b] −→ R be a twice differentiable function and assume thereexist real constants m and M such that:

m ≤ f ′′(x) ≤M for all x ∈ [a, b].

Then

(5)

∣∣∣∣f(a) + f(b)

2+ f(

a+ b

2)− 2

b− a

∫ b

af(x)dx− (b− a)[f ′(b)− f ′(a)]

24

∣∣∣∣≤ (M −m)(b− a)2

64.

Proof. Taking into account that 0 ≤(x− a+ b

2

)q(x) ≤ (b− a)2

16and m ≤ f ′′(x) ≤

M , for every x ∈ [a, b], and applying the inequality of Gruss (see [2,3]), we obtainthe following inequality:∣∣∣∣ 1

b− a

∫ b

a

(x− a+b

2

)q(x)f ′′(x)dx− 1

b− a

∫ b

a

(x− a+b

2

)q(x)dx

1

b− a

∫ b

af ′′(x)dx

∣∣∣∣≤ 1

64(M −m)(b− a)2.

This is equivalent to the inequality of the statement.

Page 72: 2014 Volume 22 No. 1

72 N. Minculete, P. Dicu, A. Ratiu

3 Applications

Application 1 We consider f(x) = xp, p > 1. Obviously f is a convex function.According to Theorem 2 one has:

ap + bp

2+

(a+ b

2

)p

− 2

b− a

bp+1 − ap+1

p+ 1≤ p(b− a)(bp−1 − ap−1)

16

so

(6) A(ap, bp) +Ap(a, b)− 2Lpp(a, b) ≤

p(b− a)(bp−1 − ap−1)

16Lp−2p−2,

where A(a, b) =a+ b

2is the arithmetic mean and Lp(a, b) =

[ap − bp

p(a− b)

] 1p−1

is the

Stolarsky′s mean.

Application 2 For f(x) =1

x, x > 0, we have f ′(x) = − 1

x2, f ′′(x) =

2

x3, hence f

is a convex function. From Theorem 2 we obtain:

1a + 1

b

2+

2

a+ b− 2(lnb− lna)

b− a≤ (b− a)

16(1

a2− 1

b2),

so

(7)1

H(a, b)+

1

A(a, b)− 2

L(a, b)≤ (b− a)2

8a2b2A(a, b),

where H(a, b) =2ab

a+ bis the harmonic mean and L(a, b) is the logarithmic mean.

Application 3 For f(x) = −lnx, x > 0, we have f ′(x) = −1

x, f ′′(x) =

1

x2> 0,

hence f is a convex function. Applying Theorem 2 for f , we find the inequality

(8) A(a, b)G(a, b)e(b−a)2

16ab ≥ I2(a, b),

where G(a, b) =√ab is the geometric mean and I(a, b) =

1

e

(bb

aa

) 1b−a

is the identric

mean.

Indeed, we have successively:

−lna− lnb

2− ln

(a+ b

2

)− 2

b− a[−b lnb+ a lna+ b− a] ≤

(b− a)[−1b +

1a ]

16

⇔ −[ln

√ab

(a+ b

2

)]+ 2 ln

(bb

aa

) 1b−a

− 2 ≤ (b− a)2

16ab

⇔ ln

1e

(bb

aa

) 1b−a

√ab(a+b

2 )≤ (b− a)2

16ab⇔(a+ b

2

)√ab e

(b−a)2

16ab ≥ 1

e

(bb

aa

) 1b−a

.

Page 73: 2014 Volume 22 No. 1

Two reverse inequalities of Bullen’s inequality 73

References

[1] S.S. Dragomir, Ch.E.M Pearce, Selected Topics on Hermite-Hadamard Inequa-lities and Applications, RGMIA Monographs, Victoria University, 2000.

[2] C.P. Niculescu, L.-E. Persson, Convex functions and their application, A con-temporary approach, Springer 2006.

[3] J. Pecaric, F. Proschan, Y.L Tong, Convex Functions, Partial Orderings andStatistical Applications, Academic Press, Inc., 1992

Nicusor MinculeteTransilvania University of BrasovDepartment of Mathematics and InformaticsStr. Iuliu Maniu, No. 50, 5091, Brasov, Romaniae-mail: [email protected]

Petrica DicuLucian Blaga University of SibiuDepartment of Mathematics and InformaticsStr. Dr. I. Ratiu, No.5-7, RO-550012 Sibiu, Romaniae-mail: [email protected]

Augusta RatiuTechnological High Scool ”Alexandru Borza”Str. Vasile Lucaciu, No 42, Cimbrud(Alba), Romaniae-mail: [email protected]

Page 74: 2014 Volume 22 No. 1
Page 75: 2014 Volume 22 No. 1

General Mathematics Vol. 22, No. 1 (2014), 75–84

The volume of the unit ball. A review 1

Valentin Gabriel Cristea

Dedicated to the late Academician Professor Dr. Dimitrie D. Stancu

Abstract

The aim of this survey is to present recent research on the problem of esti-mating of the volume of unit n-dimensional ball. Some results from the theoryof approximating of the gamma function are used. Finally, some inequalities onthe area of the unit n-dimensional ball are given.

2010 Mathematics Subject Classification: 33B15, 51M16, 51N20.

Key words and phrases: volume of the unit n-dimensional ball, surface area ofthe unit n-dimensional ball, Gamma function, monotonicity, inequalities.

1 Introduction

Inequalities on Euler’s gamma function Γ have requested the attention of manyauthors. In particular, several researchers have proven interesting monotonicityproperties of the volume of the unit ball in Rn,

(1) Ωn =πn/2

Γ(n2 + 1), n = 1, 2, 3, ... .

The volume Ωn(R) of the n-dimensional ball of radius R can be expressed using

polar coordinates (ρ, θ) by integrating Ωn−2

(√R2 − ρ2

)over the intersection disk

of (x1, x2)-plane with n-dimensional ball. Hence

Ωn(R) =2πR2

nΩn−2(R), n ≥ 3

1Received 11 June, 2014Accepted for publication (in revised form) 3 September, 2014

75

Page 76: 2014 Volume 22 No. 1

76 V.G. Cristea

and consequently

Ωn(R) =

2n2 π

n2 Rn

2·4·...·n if n is even

2n−12 π

n−12 Rn

1·3·...·n if n is odd

n2Rn

Γ(n2 + 1).

As Ωn(R) = ΩnRn, we get formula (1) for R = 1.

2 Estimates of Ωn−1

Ωn

Anderson et al. [5] and Klain and Rota [13] have presented the following inequalitiesfor every n ≥ 1 √

n

2π≤ Ωn−1

Ωn≤√n+ 1

2π,

then Alzer [2, Theorem 2] and Qi and Guo [28, Theorem 2] have exploited themonotonicity of the function

p(x) =

[Γ(x+ 1)

Γ(x+ 1

2

)]2 − x

to establish the following estimates

(2)

√n+A

2π≤ Ωn−1

Ωn≤√n+B

2π,

with the best possible constants A = 12 and B = π

2 − 1. Inequalities of type (2) werealso discussed by Borgwardt [7, p. 253], Qiu and Vuorinen [34], or Karayannakis[12, p. 2-4].

Mortici [27, Theorem 3] has improved the previous results by introducing thefollowing estimates

(3)

√n+ 1

2

2π≤ Ωn−1

Ωn≤

√n+ 1

2

2π+

1

16πn,

as a result of the fact that g(x) < 0 on (0,∞) , where

g(x) = lnΓ(x+ 1)− ln Γ

(x+

1

2

)− 1

2ln

(x+

1

4+

1

32x

).

3 Estimates of Ωn/(n+1)n+1

There are presented in Anderson et al. [5] and Klain and Rota [13] the followinginequality for every n ≥ 1

Ωn/(n+1)n+1 < Ωn.

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The volume of the unit ball. A review 77

Alzer [2, Theorem 1] has used the monotonicity of the sequence

xn = lnΩn − n

n+ 1lnΩn+1

to establish the following double inequality:

(4) aΩn/(n+1)n+1 < Ωn < bΩ

n/(n+1)n+1 , n ≥ 1,

with the best possible constants a = 2/√π = 1, 1283... and b =

√e = 1, 6487... .

F. Qi and B.-N. Guo [29], [30], [32] have rediscovered and refined Alzer’s inequa-lity (4) as a consequence of the logarithmically completely monotonicity on (−2,∞)of the function

Q(x) =

[πx/2

Γ(1 + x/2)

]1/x.

Later Mortici [27, Theorem 2] improved (4) as

(5)k

2n√2π

≤ Ωn

Ωn/(n+1)n+1

≤√e

2n√2π, n ≥ 4,

where k =64 · 72011/22 · 21/22

10395 · π5/11= 1, 5714... . Inequality in the left-hand side of (5)

holds with equality if and only if n = 11. Yin [35, Theorem 3.3] has proposed

√e

2n+2√2π

(√n+ 4

3

) 2n+1n+1√

(n+ 1)(n+ 1 + β

2

) < Ωn

Ωn/(n+1)n+1

<

√e

2n+2√2π

(√n+ 1 + β

) 2n+1n+1√

(n+ 1)(n+ 7

6

) , n ≥ 1,

where β = 3

√30130 − 2 = 0, 3533... .

4 Estimates of Ω2n

Ωn−1Ωn+1

Anderson et al. [5] and Klain and Rota [13] have presented the following inequalitiesfor every n ≥ 1

1 <Ω2n

Ωn−1Ωn+1< 1 +

1

n.

Alzer [2, Theorem 3] has found

(6)

(1 +

1

n

<Ω2n

Ωn−1Ωn+1<

(1 +

1

n

with α = 2− log2 π = 0, 34850... and β = 1/2, as a consequence of the monotonicityof the sequence

zn =2 lnΩn − lnΩn−1 − lnΩn+1

ln(n2 + 1

) .

Page 78: 2014 Volume 22 No. 1

78 V.G. Cristea

Inequalities of type (6) were also studied by Karayannakis [12, p. 5-6], or Yinwho proposed the following estimates for every integer n ≥ 1 :

(n+ 1)(n+ 1

6

)(n+ β)2

<Ω2n

Ωn−1Ωn+1<

(n+ 1)(n+ β

2

)(n+ 1

3

)2and

n+ 16

n+ β<

Ω2n

Ωn−1Ωn+1<n+ 1

n+ 13

,

where β =3

√301

30− 2 = 0, 3533... .

Mortici [27, Theorem 4] has improved (6), showing that for every integer n ≥ 4,

(7)

(1 +

1

n

) 12− 1

4n

<Ω2n

Ωn−1Ωn+1<

(1 +

1

n

) 12

.

5 Estimates ofΩ

1/(n+1)n+1

Ω1/nn

andΩ

1/(n+2)n+2

Ω1/nn

Qi and Guo [29] and Qi and Guo [31] have proved the following double inequalities√n+ 2

n+ 3<

Ω1/(n+1)n+1

Ω1/nn

< 4

√n+ 2

n+ 3√n+ 2

n+ 4<

Ω1/(n+2)n+2

Ω1/nn

< 4

√n+ 2

n+ 4,

respective

1

π2/(n−2)n

√n+ 2

n+ 4<

Ω1/(n+2)n+2

Ω1/nn

<1

π2/(n−2)n8

√n+ 2

n+ 4.

Qi and Guo [29, Remark 9] have proposed as an open problem the finding of thebest positive constants a ≥ 3, b ≤ 3, λ ≤ 1, µ ≥ 1, α ≥ 2 and β ≤ 4 such that thedouble inequality holds true for every n ≥ 1 :

α

√1− λ

n+ a<

Ω1/(n+1)n+1

Ω1/nn

< β

√1− µ

n+ b.

Yin [35, Theorem 3.1] has proved that

Ωn ≤ (Ω1Ω2...Ωn)1

n−1 , n = 1, 2, 3, ...

and

(Ω1Ω2...Ωn)1n ≤ Ω(n+1)/2, n = 1, 3, 5, ... .

Page 79: 2014 Volume 22 No. 1

The volume of the unit ball. A review 79

Let Hn be the n-th harmonic number. In [35, Theorem 3.4], Yin has proved that

the sequence (Ωn)1

Hn n≥1 attains its maximum at n = 3, while the sequence

(Ωn)1

Hn n≥3 is monotonically decreasing to zero. Further similar results wereobtained by Qi and Guo in [33, Theorem 1-2] by exploiting the properties of thefollowing functions defined on (0,∞)r 1 :

(8) F (x) =lnΓ(x+ 1)

ln(x2 + 1)− ln (x+ 1)

and

(9) G(x) =

[πx

Γ (x+ 1)

]1/[ln(x2+1)−ln(x+1)].

As a consequence of the fact that (8)-(9) are monotone, the sequence

Ω1/[ln(n2/4+1)−ln(n/2+1)]n is strictly decreasing for n ≥ 3 and logarithmically convex

for n ≥ 1.Bhayo [6, Theorem 1.4] has proved the following inequalities for every integers

k, n ≥ 1: √Ω2nΩ2(n−1) ≤ Ω(2n−1)

and (Ωn−1

Ωn

)k

≤Ωk(n−1)

Ωkn.

Anderson and Qiu [4, Conjecture 3.3] have conjectured that the function

(10)ln Γ(x+ 1)

x ln(x)

is concave on (0,∞). Chen, Qi and Li [9], [14] have proven that function (10) isstrictly increasing on (0,∞), while Elbert and Laforgia [10, Section 3] have proventhat (10) is concave on x ∈ (1,∞) . Alzer has demonstrated in [2]-[3] that the function

(11) F (x) =lnΓ(x+ 1)

x ln(2x)

is strictly increasing on [1,∞) and strictly concave on [46,∞). As a consequence,he has proposed the following double inequality

exp

(a

n (lnn)2

)≤ Ω

1/(n lnn)n

Ω1/[(n+1) ln(n+1)]n+1

< exp

(b

n (lnn)2

),

valid for every n ≥ 2 if and only if a ≤ ln 2 lnπ − 2 (ln 2)2 ln (4π/3)

3 ln 3= 0, 3... and

b ≥ 1 + ln (2π)

2= 1, 4.... Qi and Guo have proved in [29] that function (11)

is strictly increasing and concave on (12 ,∞), then they have concluded that the

sequence (Ω1/(n lnn)n )n≥2 is strictly logarithmically convex. These results solve and

generalize the conjecture in [5, Remark 2.41] and the result in [4, Corrolary 3.1]1997.

Page 80: 2014 Volume 22 No. 1

80 V.G. Cristea

6 A new estimate for ratio of Ωn

In this final section we show a general method that can be used to improve aboveresults. In order to illustrate this method we present the following refinement ofinequalities (3).Theorem 1. The following inequalities are valid for every integer n ≥ 2:

(12)

√n+ 1

2

2π+

1

16πn− 1

32πn2− 5

256πn3≤ Ωn−1

Ωn≤

√n+ 1

2

2π+

1

16πn− 1

32πn2.

In the proof we use the following well-known inequalities

(13) u(x) < Γ(x+ 1) < v(x)

where

u(x) = ln√2π +

(x+

1

2

)lnx− x+

1

12x− 1

360x3

and

v(x) = ln√2π +

(x+

1

2

)lnx− x+

1

12x− 1

360x3+

1

1260x5.

We need the following lemma:Lemma 1. The following inequalities are valid for every x ≥ 3 in the left-hand sideand x ≥ 5 in the right-hand side:

(14)

√1

4+

1

2x+

1

16x− 1

32x2− 5

256x3<

Γ(x2 + 1

)Γ(x−12 + 1

) <√1

4+

1

2x+

1

16x− 1

32x2

Proof. For the right-hand side inequality (14), we have to prove that g(x) < 0,where

g(x) = lnΓ(x+ 1)− ln Γ

(x+

1

2

)− 1

2ln

(x+

1

4+

1

32x− 1

128x2

).

Using (13), we obtain g(x) < r(x),where

r(x) = v(x)− u

(x− 1

2

)− 1

2ln

(x+

1

4+

1

32x− 1

128x2

)and it suffices to show that r(x) < 0. We have

r′′(x) = − S(x)

210x7 (2x− 1)5 (4x+ 32x2 + 128x3 − 1)2

where S(x) = 2688 000x12 + ... is a polynomial of degree 12. As the polynomialS (x− 5) has all positive coefficients, it results that S(x) > 0 on (5,∞) .

Now r is strictly concave on (5,∞) with r (∞) = 0, so r(x) < 0. Hence g(x) < 0and the first part of the proof is finished.

Page 81: 2014 Volume 22 No. 1

The volume of the unit ball. A review 81

For left-hand side in (14), we have to prove that h(x) > 0, where

h(x) = lnΓ(x+ 1)− ln Γ

(x+

1

2

)− 1

2ln

(x+

1

4+

1

32x− 1

128x2− 5

2048x3

).

Using (13), we get h(x) > t(x), where

t(x) = u(x)− v

(x− 1

2

)− 1

2ln

(x+

1

4+

1

32x− 1

128x2− 5

2048x3

).

and it suffices to prove that t(x) > 0. We have

t′′(x) =T (x)

210x5 (2x− 1)7 (64x2 − 16x+ 512x3 + 2048x4 − 5)2

where T (x) = 2063 728 640x13 + ... is a polynomial of degree 13. As the polynomialT (x− 3) has all coefficients positive, we deduce that T (x) > 0 on (3,∞) .

Now t is strictly convex on (3,∞) with t (∞) = 0, so t(x) > 0 and the proof iscompleted.The proof of Theorem 1. By direct computation, inequalities (12) hold true forn = 2, 3, 4. Finally, (14) follows by multiplying (12) by

√π.

The surface area of the unit ball in n dimensions can be expressed as

An =nπn/2

Γ(n2 + 1), n = 1, 2, 3, ...,

or An = nΩn. Mortici [27, Theorems 2–4] has obtained the following inequalities onAn:

nk2n√2π (n+ 1)

nn+1

≤ An

An/(n+1)n+1

≤ n√e

2n√2π (n+ 1)

nn+1

(n ≥ 4)

(n− 1)

√1

2πn+

1

4πn2≤ An−1

An≤ (n− 1)

√1

2πn+

1

4πn2+

1

16πn3(n ≥ 1)(

1− 1

n

)(1 +

1

n

) 12

<An−1An+1

A2n

<

(1 +

1

n

) 12− 1

4n

(n ≥ 4).

Using Theorem 1, we can state the following estimates on the surface area An of theunit n-dimensional ball.Corollary 1. For every n ≥ 2 we have the following inequalities:

(n− 1)

√1

2πn+

1

4πn2+

1

16πn3− 1

32πn4− 5

256πn5≤ An−1

An

≤ (n− 1)

√1

2πn+

1

4πn2+

1

16πn3− 1

32πn4.

Acknowledgements. The author thanks Prof. Cristinel Mortici for suggestingthe problem and for his guidance through out the progress of this work. Somecomputations made in this paper were performed using Maple software.

Page 82: 2014 Volume 22 No. 1

82 V.G. Cristea

References

[1] H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp.,vol. 66, no. 217, 1997, 373-389.

[2] H. Alzer, Inequalities for the volume of the unit ball in Rn, J. Math. Anal.Appl., no. 252, 2000, 353-363.

[3] H. Alzer, Inequalities for the volume of the unit ball in Rn, Mediterr. J. Math.,vol. 5, no. 4, 2008, 395–413.

[4] G. D. Anderson, S.-L. Qiu, A monotonicity property of the gamma function,Proc. Amer. Math. Soc., vol. 125, 1997, 3355-3362.

[5] G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen, Special functions of qua-siconformal theory, Expo. Math., no. 7, 1989, 97-136.

[6] B. A. Bhayo, On functional inequalities for the psi function, arXiv:1210.0046v1[math.CA] 28 Sep 2012.

[7] K. H. Borgwardt, The Simplex Method, Springer Berlin (1987).

[8] J. Bohm, E. Hertel, Polyedergeometrie in n-dimensionalen Raumen konstanterKrummung, Birkhauser Basel (1981).

[9] Ch.-P. Chen, F. Qi, Note on a monotonicity property of the gamma function,Octogon Math. Mag., vol. 12, no. 1, 2004, 123–125.

[10] A. Elbert, A. Laforgia, On some properties of the gamma function, Proc. Amer.Math. Soc., vol. 128, no. 9, 2000, 2667–2673.

[11] B.-N. Guo, F. Qi, A class of completely monotonic functions involving divideddifferences of the psi and tri-gamma functions and some applications, J. KoreanMath. Soc., vol. 48, no. 3, 2011, 655-667.

[12] D. Karayannakis, Gautchi’s ratio and the volume of the unit ball in Rn. Avail-able online at http://arxiv.org/abs/0902.2182.

[13] D. A. Klain, G.-C. Rota, A continuous analogue of Sperner’s theorem, Commun.Pure Appl. Math., vol. 50, 1997, 205–223.

[14] X. Li, Monotonicity properties for the gamma and psi functions, Sci. Magna,vol. 4, no. 4, 2008, 18–23.

[15] C. Mortici, Product approximations via asymptotic integration, Amer. Math.Monthly, vol. 117, no. 5, 2010, 434-441.

[16] C. Mortici, An ultimate extremely accurate formula for approximation of thefactorial function, Arch. Math. (Basel), vol. 93, no. 1,2009, 37–45.

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The volume of the unit ball. A review 83

[17] C. Mortici, New approximations of the gamma function in terms of the digammafunction, Appl. Math. Lett., vol. 23, no. 1, 2010, 97–100.

[18] C. Mortici, New sharp bounds for gamma and digamma functions, An. Stiint.Univ. A. I. Cuza Iasi Ser. N. Matem., vol. 57, no. 1, 2011, 57-60.

[19] C. Mortici, Complete monotonic functions associated with gamma function andapplications, Carpathian J. Math., vol. 25, no. 2, 2009, 186–191.

[20] C. Mortici, Sharp inequalities related to Gosper’s formula, Comptes RendusMath. Acad. Sci. Paris, Ser. I, vol. 348, 2010, 137-140.

[21] C. Mortici, A class of integral approximations for the factorial function, Com-put. Math. Appl., vol. 59, no. 6, 2010, 2053-2058.

[22] C. Mortici, Best estimates of the generalized Stirling formula, Appl. Math. Com-put., vol. 215, no. 11, 2010, 4044– 4048.

[23] C. Mortici, Improved convergence towards generalized Euler–Mascheroni con-stant, Appl. Math. Comp., vol. 215, no. 9, 2010, 3443–3448.

[24] C. Mortici, Sharp inequalities and complete monotonicity for the Wallis ratio,Bull. Belgian Math. Soc. Simon Stevin, vol. 17, 2010, 929-936.

[25] C. Mortici, Very accurate estimates of the polygamma functions, Asympt. Anal.,vol. 68, no. 3, 2010, 125-134.

[26] C. Mortici, Optimizing the rate of convergence in some newclasses of sequencesconvergent to Euler’s constant, Anal. Appl. (Singap.), vol. 8, no. 1, 2010, 1–9.

[27] C. Mortici, Monotonicity properties of the volume of the unit ball in Rn, Opti-mization Lett., vol. 4, 2010, 457-464.

[28] F. Qi, B.-N. Guo, A class of completely monotonic functions involving di-vided differences of the psi and polygamma functions and some applications,arXiv:0903.1430v1 [math.CA] 8 Mar 2009.

[29] F. Qi, B.-N. Guo, Monotonicity and logarithmic convexity relating to the volumeof the unit ball, arXiv:0902.2509v2 [math.CA] 7 Sep 2010.

[30] F. Qi, B.-N. Guo, Complete monotonicities of functions involving the gammaand digamma functions, RGMIA Res. Rep. Coll., vol. 7, no. 1, Art. 8, 2004,63–72.

[31] F. Qi, B.-N. Guo, Complete monotonicity of a function involving a ratio ofgamma functions and applications, arXiv:0904.1101v3 [math.CA] 25 Jan 2011.

[32] F. Qi, B.-N. Guo, Some logarithmically completely monotonic functions relatedto the gamma function, J. Korean Math. Soc., vol. 47, no. 6, 2010, 1283–1297.

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84 V.G. Cristea

[33] F. Qi, B.-N. Guo, Two monotonic functions involving gamma function andvolume ball, Available online at http://arxiv.org/abs/1001.1496.

[34] S.-L. Qiu, M. Vuorinen, Some properties of the gamma and psi functions, withapplications, Math. Comp., vol. 74, 2004, 723–742.

[35] L. Yin, Several inequalities for the volume of the unit ball in Rn,Availableonline at http://www.emis.de/journals/BMMSS/pdf/acceptedpapers/2012-05-010-R2.pdf.

[36] Y. Yu, An inequality for ratios of gamma functions, J. Math. Anal. Appl. vol.352, 2009, 967–970.

Valentin Gabriel CristeaPh. D. Student, University Politehnica of BucharestSplaiul Independenei 313, Bucharest, Romaniae-mail: [email protected]

Page 85: 2014 Volume 22 No. 1

General Mathematics Vol. 22, No. 1 (2014), 85–90

Ramanujan type formulas for approximating thegamma function. A survey 1

Sorinel Dumitrescu

Dedicated to the late Academician Professor Dr. Dimitrie D. Stancu

Abstract

In this survey we discuss Ramanujan formula and related formulas forapproximating the gamma function as many improvements were presented inthe recent past. In the final part some new inequalities are presented.

2010 Mathematics Subject Classification: 33B15, 26D15, 11Y25, 41A25,34E05.

Key words and phrases: Gamma function, approximations, Ramanujanformula.

1 Recent improvements on Ramanujan’s formula

In mathematics and science in general, we usually face with situations in which weare forced to approximate large factorials. One of the most known formula used toestimate big factorials and gamma function is the Stirling formula

Γ (x+ 1) ∼√2πx

(xe

)x.

However, in pure mathematics more accurate approximations are required. A betterestimate of the factorial function is given by Gosper’s formula

Γ (x+ 1) ∼√2π(xe

)x√x+

1

6,

or Ramanujan’s formula

Γ (x+ 1) ∼√2π(xe

)x6

√x3 +

1

2x2 +

1

8x+

1

240.

1Received 11 June, 2014Accepted for publication (in revised form) 30 July, 2014

85

Page 86: 2014 Volume 22 No. 1

86 S. Dumitrescu

which is a consequence of the formula

Γ(x+ 1) ∼√π(xe

)x6

√8x3 + 4x2 + x+

ϑx30

with limx→∞

ϑx = 1.

See [10], [1]-[9] and all references therein. Also, in [10] there have been establishedthe following inequalities for every x ≥ 1 :(1)√π(xe

)x6

√8x3 + 4x2 + x+

1

100< Γ(x+ 1) <

√π(xe

)x6

√8x3 + 4x2 + x+

1

30.

Motivated by these inequalities, Anderson, Vamanamurthy and Vuorinen [1]defined the function

h(x) =

Γ(x+ 1)√

π

( ex

)x6

−(8x3 + 4x2 + x

)and asked whether h(x) is increasing from (1,∞) into

(1

100,1

30

). Karatsuba [4]

answered affirmatively to this question then she presented the following improvementof (1):

√π(xe

)x6

√8x3 + 4x2 + x+

e6

π3− 13 ≤ Γ(x+ 1) <

√π(xe

)x6

√8x3 + 4x2 + x+

1

30.

These inequalities are sharp, as limx→∞

h(x) =1

30and h(1) =

e6

π3− 13.

Mortici [5] introduced the new approximation formula:

Γ (x+ 1) ∼√π(xe

)x4

√4x2 +

4

3x+

2

9

and proved that the associated function

H(x) =

Γ(x+ 1)√

π

( ex

)x4

−(4x2 +

4

3x

)

is strictly increasing on [1,∞) from H (1) = α :=e4

π2− 16

3= 0.198 · · · to

H (∞) = β :=2

9= 0.222 · · · . The double inequality H (1) ≤ H (x) < H (∞)

can be written as

√π(xe

)x4

√4x2 +

4

3x+ α ≤ Γ(x+ 1) <

√π(xe

)x4

√4x2 +

4

3x+ β.

The following new formulas are also presented in [5]:

Γ (x+ 1) ∼√2π(xe

)x8

√x4 +

2

3x3 +

2

9x2 +

11

405x− 8

1215

Page 87: 2014 Volume 22 No. 1

Ramanujan type formulas for approximating the gamma function. A survey 87

and

Γ (x+ 1) ∼√2π(xe

)x10

√x5 +

5

6x4 +

25

72x3 +

89

1296x2 − 95

31104x+

2143

1306368

that are much stronger than Ramanujan formula (3).The general method for establishing increasingly more accurate formulas of order

2kΓ (x+ 1) ∼

√2π(xe

)x2k√xk + · · ·

is also presented by Mortici in [6]. The idea is to start from the standard series

Γ (x+ 1) ∼√2πx

(xe

)xexp

(1

12x− 1

360x3+ · · ·

)=

√2πx

(xe

)xexp

[2k

1

12x− 1

360x3+ · · ·

] 12k

and to use the transformation exp t = 1 + t/1! + t2/2! + · · · .This result was sistematically proven by Chen and Lin in [2].We also mention the following class of approximations presented very recent by

Dumitrescu and Mortici in [3]

Γ (x+ 1) ∼√2πx

(xe

)x6

√√√√1 +1

2(x+ 2a− 1

4

) + a(x+ 2a− 1

4

)2 +2a2 − 13

480(x+ 2a− 1

4

)3that gives better results than Ramanujan’s formula.

Other improvement was presented by Mortici in [6], who proposed the followingformula of Ramanujan type, starting from Burnside’s formula:

(2) Γ (x+ 1) ∼√

e

(x+ 1

2

e

)x

6

√x3 +

5

4x2 +

17

32x+

173

1920.

Moreover, he proved that the following inequalities are valid for every x ≥ 13:√2π

e

(x+ 1

2

e

)x

6

√x3 +

5

4x2 +

17

32x+

172

1920

< Γ(x+ 1)

<

√2π

e

(x+ 1

2

e

)x

6

√x3 +

5

4x2 +

17

32x+

173

1920.

2 New Results

Motivated by the Ramanujan-Burnside formula (2), we propose in this section thefollowing family of approximations as x→ ∞:

(3) Γ (x+ 1) ∼ σ (x) :=

√2πx

(1 +

a

x4

)(xe

)x6

√1 +

1

2x+

1

8x2+

1

240x3,

Page 88: 2014 Volume 22 No. 1

88 S. Dumitrescu

where a is any real number.By using a method first introduced by Mortici in [7], we define the relative error

sequence Γ(n+ 1) = σ(n) · exp wn . As

wn − wn+1 = −(2a+

11

2880

)1

n5+

(5a+

29

2016

)1

n6+O

(1

n7

),

the best estimate is obtained when 2a+ 112880 = 0, that is a = − 11

5760 .In the following table we prove the superiority of our new formula (3) over

Ramanujan’s formula.

n ρ(n)− Γ(n+ 1) Γ(n+ 1)− σ(n)

10 0.311 61 3. 488 7× 10−2

50 4. 552 5× 1054 9. 411 8× 1052

100 8. 821 1× 10146 9. 026 7× 10144

500 1. 860 3× 101120 3. 776 6× 101117

1000 3. 838 3× 102552 3. 892 1× 102549

Moreover, further computations we made show us that formula (3) is much betterthan Burnside-Ramanujan formula (2).

Finally we give the following estimates associated to our new formula (3).

Theorem 1 The following inequalities are valid for every x ≥ 2 :√2πx

(1− 11

5760x4

)(xe

)x6

√1 +

1

2x+

1

8x2+

1

240x3< Γ (x+ 1)

<

√2πx

(1− 11

5760x4

)(xe

)x6

√1 +

1

2x+

1

8x2+

1

240x3· exp

13

13 44x5

We prove that F > 0 and G < 0 on [2,∞), where

F (x) = lnΓ (x+ 1)√

2πx

(1 +

−115760x4

)(xe

)x 6

√1 + 1

2x + 18x2 + 1

240x3

and

G(x) = lnΓ (x+ 1)√

2πx

(1 +

−115760x4

)(xe

)x 6

√1 + 1

2x + 18x2 + 1

240x3 · exp

1313 44x5

.

We have

F ′(x) = ψ(x)− lnx+1

2x+

11

2880x5(

115760x4 − 1

) + 1

6

12x2 + 1

4x3 + 180x4

12x + 1

8x2 + 1240x3 + 1

Page 89: 2014 Volume 22 No. 1

Ramanujan type formulas for approximating the gamma function. A survey 89

and

G′(x) = ψ(x)− lnx+1

2x+

11

2880x5(

115760x4 − 1

) + 1

6

12x2 + 1

4x3 + 180x4

12x + 1

8x2 + 1240x3 + 1

+65

1344x6,

where ψ = Γ′/Γ is the digamma function. By using the standard inequalities

− 1

12x2+

1

120x4− 1

252x6+

1

240x8− 5

660x10< ψ (x)− lnx+

1

2x

< − 1

12x2+

1

120x4− 1

252x6+

1

240x8,

we get F ′ < u and G′ > v, where

u(x) = − 1

12x2+

1

120x4− 1

252x6+

1

240x8+

11

2880x5(

115760x4 − 1

)+1

6

12x2 + 1

4x3 + 180x4

12x + 1

8x2 + 1240x3 + 1

and

v(x) = − 1

12x2+

1

120x4− 1

252x6+

1

240x8− 5

660x10+

11

2880x5(

115760x4 − 1

)+1

6

12x2 + 1

4x3 + 180x4

12x + 1

8x2 + 1240x3 + 1

+65

1344x6.

As rational functions, simple computations using Maple software show us that u < 0and v > 0 on [2,∞). In consequence, F ′ < 0 and G′ > 0 on [2,∞). Finally, takinginto account the obtained monotonicity of F and G with F (∞) = G (∞) = 0, wededuce that F > 0 and G < 0 and the proof is now completed.

Acknowledgements. The author thanks Prof. Cristinel Mortici for indications inwriting of this paper. Some computations made in this paper were performed usingMaple software.

References

[1] G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen, Conformal Invariants,Inequalities, and Quasiconformal Maps, Can. Math. Soc., Ser. Mon. Adv. Txt.Wiley, New York, 1997.

[2] C.-P. Chen, L. Lin, Remarks on asymptotic expansions for the gamma function,Appl. Math. Lett., vol. 25, 2012, 2322-2326.

[3] S. Dumitrescu, C. Mortici, Refinements of Ramanujan formula for gamma func-tion, Intern. J. Pure Appl. Math., 2014, in press.

[4] E. A. Karatsuba, On the asymptotic representation of the Euler gamma functionby Ramanujan, J. Comput. Appl. Math., vol. 135, 2001, 225–240.

Page 90: 2014 Volume 22 No. 1

90 S. Dumitrescu

[5] C. Mortici, On Ramanujan’s large argument formula for the Gamma function,Ramanujan J., vol. 26, no. 2, 2011, 185-192.

[6] C. Mortici, Ramanujan formula for the generalized Stirling approximation, Appl.Math. Comp., vol. 217, no. 6, 2010, 2579-2585.

[7] C. Mortici, Product approximations via asymptotic integration, Amer. Math.Monthly, vol. 117, no. 5, 2010, 434-441.

[8] C. Mortici, An improvement of the Ramanujan formula for approximation of theEuler Gamma function, Carpathian J. Math., vol. 28, no. 2, 2012, 285-287.

[9] C. Mortici, Ramanujan’s estimate for the gamma function via monotonicityarguments, Ramanujan J., vol. 25, no. 2, 2011, 149-154.

[10] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa,Springer, New Delhi, Berlin. Introduction by G.E. Andrews, 1988.

Sorinel DumitrescuPh. D. Student, University Politehnica of BucharestSplaiul Independenei 313, Bucharest, Romaniae-mail: [email protected]

Page 91: 2014 Volume 22 No. 1

General Mathematics Vol. 22, No. 1 (2014), 91–98

Approximation of fractional derivatives by Bernsteinoperators 1

Radu Paltanea

Dedicated to the late Academician Professor Dr. Dimitrie D. Stancu

Abstract

We show the property of uniform approximation of fractional derivatives inCaputo sense by using Bernstein operators and give estimates with the first andthe second order moduli for the degree of approximation.

2010 Mathematics Subject Classification: 41A36, 41A25, 41A28, 26A33

Key words and phrases:Bernstein operators, second order modulus, fractionalderivative, simultaneous approximation, Kantorovich type operators

1 1. Introduction

The fractional differential calculus is a classical subject of Analysis with many practi-cal applications in different areas like mechanics, biochemistry, electrical engineering,medicine etc., see [3]. The theoretical study of fractional derivatives is connectedwith the theory of fractional differential and integral equations. In approximationtheory there exist several directions of investigation. We mention only the recentresults concerning the approximation by linear positive operators of Anastassiou [1],[2].

In our paper we consider another problem, namely the approximation of frac-tional derivatives by the classical Bernstein operators. In this section we give nota-tions and we recall some basic facts of fractional calculus. In Section 2 we obtaincertain auxiliary results about Bernstein operators and in Section 3 we give the mainresults.

1Received 24 June, 2014Accepted for publication (in revised form) 29 August, 2014

91

Page 92: 2014 Volume 22 No. 1

92 R. Paltanea

Let I be an interval. For α > 0 and a ∈ I, the Riemann-Liouville operator isgiven by

(1) Jαa (f, x) =

1

Γ(α)

∫ x

af(t)(x− t)α−1dt,

where f is integrable on I and x ∈ I, x > a. For α = 0, denote J0a = I, identity

operator. If a = 0 ∈ I denote Jα instead of Jα0 . We have

Jα Jβ = Jα+β, α, β ≥ 0.

For p ∈ N0 = 0, 1, 2, . . ., denote also by Dp the usual derivative operator of orderp.

The two main extensions of the derivative operator for a non-integer order α ≥ 0are the derivatives in Liouville-Riemann sense and the derivative in Caputo sense.These are defined as follows. Denote by ⌈α⌉ the smallest integer that is greater thanα.

a. Fractional derivative in Riemann-Liouvile sense

For α > 0, let p ∈ N, p ≥ ⌈α⌉ and a ∈ I, define

Dαa = Dp Jp−α

a .

For α = 0 define D0a = I, the identity operator. For a = 0 denote simple Dα. The

definition of Dαa does not depend on the choice of p.

b. Fractional derivative in Caputo sense

For α ≥ 0, let p ∈ N, p = ⌈α⌉ and a ∈ I, define

Dαa = Jp−α

a Dp.

For a = 0 denote simple Dα.

Remark 1 Let a ∈ I. We have

Dαa = Dα = Dα

a , if α ∈ N.

Between the two definitions there is the following connection:

Theorem A Let α ≥ 0 and let p = ⌈α⌉. Then, for a ∈ I and any functionf ∈ ACp−1(I) we have almost everywhere

Dαa f = Dα

a (f − Tp−1[f ; a]),

where Tp−1[f ; a] denotes the Taylor polynomial of degree p−1 of function f , centeredin a.

If I is an interval we denote by F(I), the linear space of real functions definedon I. For s ≥ 0 define es(t) = ts, t ∈ [0, 1]. Also, for n ∈ N and p ∈ N denote(n)p := n(n− 1) . . . (n− p+ 1).

Page 93: 2014 Volume 22 No. 1

Approximation of fractional derivatives by Bernstein operators 93

2 Auxiliary results for Bernstein operators

The Bernstein operators are given by:

Bn(f, x) =

n∑k=0

pn,k(x)f

(k

n

), f ∈ R[0,1], x ∈ [0, 1], n ∈ N,

where

pn,k(x) =n!

k!(n− k)!xk(1− x)n−k.

It is well known that the usual derivatives of Bernstein operators may be expressedin the following form:Theorem B For n ∈ N, p ∈ N, f ∈ Cp[0, 1] and x ∈ [0, 1] we have,

B(p)n (f, x) = (n)p

n−p∑k=0

pn−p,k(x)

∫ 1n

0. . .

∫ 1n

0f (p)(

k

n+ u1 + . . .+ up)du,

where du := du1 . . . dupFor n, p ∈ N consider the linear positive operator Ln,p : C[0, 1] → C[0, 1], given

by:

Ln,p(g, x) = (n)p

n−p∑k=0

pn−p,k(x)

∫ 1n

0. . .

∫ 1n

0g(k

n+ u1 + . . .+ up)du,

g ∈ C[0, 1], x ∈ [0, 1].

Lemma 1 For any n ∈ N, p ∈ N and x ∈ [0, 1] we have:

i) Ln,p(e0, x) =(n)pnp ,

ii) Ln,p(e1, x) =(n)pnp

[n−pn x+ p

2n

],

iii) Ln,p(e2, x) =(n)pnp

[(n−p)(n−p−1)

n2 · x2 + (n−p)(p+1)n2 · x+ 3p2+p

12n2

].

Proof.

i) Relation Ln,p(e0, x) =(n)pnp is immediate.

ii) We have

Ln,p(e1, x) = (n)p

n−p∑k=0

pn−p,k(x)

∫ 1n

0. . .

∫ 1n

0

(kn+ u1 + . . .+ up

)du

= (n)p

n−p∑k=0

pn−p,k(x)[kn· 1

np+p

2

( 1n

)p+1]=

(n)pnp

[n− p

nx+

p

2n

].

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94 R. Paltanea

iii) We have

Ln,p(e2, x) = (n)p

n−p∑k=0

pn−p,k(x)

∫ 1n

0. . .

∫ 1n

0

(kn+ u1 + . . .+ up

)2du

= (n)p

n−p∑k=0

pn−p,k(x)

∫ 1n

0. . .

∫ 1n

0

[(kn

)2+ 2

p∑j=1

k

nuj

+

p∑j=1

(uj)2 + 2

∑1≤i<j≤p

uiuj

]du

= (n)p

n−p∑k=0

pn−p,k(x)[(kn

)2 1

np+ 2p · k

n· 1

2nn+1+

+p

3· 1

np+2+ 2 · 1

4

(p

2

)1

np+2

]=

(n)pnp

[(n− p)(n− p− 1)

n2· x2 + (n− p)(p+ 1)

n2· x+

3p2 + p

12n2

].

Corollary 1 For any n ∈ N, p ∈ N and x ∈ [0, 1] we have:

i) Ln,p(e1 − xe0, x) =(n)pnp · p

2n(1− 2x),

ii) Ln,p((e1 − xe0)2, x) =

(n)pnp

[n−p2−p

n2 x(1− x) + 3p2+p12n2

].

Corollary 2 For any n ∈ N and p ∈ N we have:

i) ∥L(e0)− e0∥ ≤ p(p−1)2n ,

ii) ∥Ln,p(e1 − •)∥ = p2n ,

iii) ∥Ln,p((e1 − •)2)∥ ≤ 14n .

3 Approximation of fractional derivatives in Caputo’ssense

We need the following simple result

Lemma 2 For each s ≥ 0, α > 0 and x ∈ [0, 1] we have

(2) Jα(es, x) =Γ(s+ 1)

Γ(α+ s+ 1)· xs+α.

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Approximation of fractional derivatives by Bernstein operators 95

Proof. From relation (1), for a = 0 and using the change of variable t = xu we have

Jα(es, x) =1

Γ(α)

∫ x

0ts(x− t)α−1dt

=1

Γ(α)

∫ 1

0xs+αus(1− u)α−1du

=xα+s

Γ(α)·B(s+ 1, α)

=Γ(s+ 1)

Γ(α+ s+ 1)· xs+α.

First we obtain estimates with the first order modulus of continuity, ω1. We usethe following general result of Mond [4], which is a variant of the estimate of Shishaand Mond.

Theorem C Let I be an arbitrary interval and let V ⊂ C(I) be a linear subspacesuch that ej ∈ V , for j = 0, 1, 2. If L : V → F(I) is a positive linear operator, thenwe have

|L(f, x)− f(x)| ≤ |f(x)| · |L(e0, x)− 1|

+(L(e0, x) +

1

h2L((e1 − xe0)

2, x))ω1(f, h),

for any f ∈ V , any x ∈ I and any h > 0.

Theorem 1 For α ≥ 0, p = ⌈α⌉, f ∈ Cp[0, 1], h > 0, n ∈ N we have

∥DαBn(f)− Dαf∥ ≤ p(p− 1)

n∥f (p)∥+

(1 +

1

4nh2

)ω1(f

(p), h).

For h = 1√nwe have

(3) ∥DαBn(f)− Dαf∥ ≤ p(p− 1)

n∥f (p)∥+ 5

4ω1

(f (p),

1√n

).

For h = 1√nand 0 ≤ α < 1 we have:

(4) ∥DαBn(f)− Dαf∥ ≤ 5

4ω1

(f ′,

1√n

).

Proof. Since (Bn)(p)(f) = Ln,p(f

(p)), using Theorem C and Corollary 2 we obtain

|(Bn)(p)(f, t)− f (p)(t)| = |Ln,p(f

(p), t)− f (p)(t)|

≤ p(p− 1)

n∥f (p)∥+

(1 +

1

4nh2

)ω1(f

(p), h).

Page 96: 2014 Volume 22 No. 1

96 R. Paltanea

Since Jp−α is positive operator, using Lemma 2 we obtain for x ∈ [0, 1]:

|DαBn(f, x)− Dαf(x)| = |Jp−α((Bn)(p)(f)− f (p), x)|

≤ Jp−α(|(Bn)(p)(f)− f (p)|, x)

≤ Jp−α(∥(Bn(f))(p) − f (p)∥e0, x)

≤ p(p− 1)

n∥f (p)∥+

(1 +

1

4nh2

)ω1(f

(p), h).

The particular cases follows immediately.

Corollary 3 Let α ≥ 0, and f ∈ Cp[0, 1], where p = ⌈α⌉. Then we have

(5) limn→∞

∥DαBn(f)− Dαf∥ = 0.

A more refined estimate can be given using a combination of first and secondorder moduli of continuity. For this we shall use the following general estimate withoptimal constants, given in Paltanea [5] or [6]. If I is an interval and h > 0 we writeh ≼ 1

2 length(I) if h ≤ 12 length(I) and I is closed of if h < 1

2 length(I) and I is notclosed.Theorem D Let I be an arbitrary interval and let V ⊂ C(I) be a linear subspacesuch that ej ∈ V , for j = 0, 1, 2. If L : V → F(I) be a positive linear operator, thenwe have

|L(f, x)− f(x)| ≤ |f(x)| · |L(e0, x)− 1|+ |L(e1 − xe0, x)|h−1ω1(f, h)

+(L(e0, x) +

1

2h2F ((e1 − xe0)

2, x))ω2(f, h),(6)

for any f ∈ V , any x ∈ I and any 0 < h ≼ 12 length(I). First we give a separate

estimate for the case of derivatives of integer orders

Theorem 2 For n ∈ N, p ∈ N, f ∈ Cp[0, 1], x ∈ [0, 1], 0 < h ≤ 12 , we have∣∣∣B(p)

n (f, x)− f (p)(x)∣∣∣ ≤ p(p− 1)

n|f (p)(x)|+ p

2n

|1− 2x|h

· ω1(f(p), h)

+

[1 +

1

2h2

(n− p− p2

n2· x(1− x) +

3p2 + p

12n2

)]ω2(f

(p), h).(7)

Also, for n ∈ N, n ≥ 4, p ∈ N and f ∈ Cp[0, 1] we have

∥B(p)n (f)− f (p)∥ ≤ p(p− 1)

n∥f (p)∥+ p

2√n· ω1

(f (p),

1√n

)+9

8· ω2

(f (p),

1√n

).(8)

Proof. We apply Theorem D for the operator L = Ln,p and take into account(Bn)

(p)(f) = Ln,p(f(p)) and Corollaries 1, 2. Relation (8) follows from relation (7)

choosing h = 1√n.

Page 97: 2014 Volume 22 No. 1

Approximation of fractional derivatives by Bernstein operators 97

Theorem 3 For α ≥ 0, p = ⌈α⌉, f ∈ Cp[0, 1], n ∈ N, n ≥ 4 we have

∥DαBn(f)− Dαf∥ ≤ p(p− 1)

n∥f (p)∥+ p

2√n· ω1

(f (p),

1√n

)+9

8ω2

(f (p),

1√n

).(9)

For α ∈ [0, 1) we obtain

∥DαBn(f)− Dαf∥ ≤ 1

2√nω1

(f ′,

1√n

)+

9

8ω2

(f ′,

1√n

).(10)

Proof. Similarly as in the proof of Theorem 1 using (8) we have successively, forx ∈ [0, 1].

|DαBn(f)(x)− Dαf(x)| ≤ Jp−α(∥(Bn(f))(p) − f (p)∥e0, x)

≤ p(p− 1)

n∥f (p)∥+ p

2√n· ω1

(f (p),

1√n

)+

9

8· ω2

(f (p),

1√n

).

The particular case 0 ≤ α < 1 follows immediately.In the last part we obtain estimates using a combination of moduli ω1

(f (p), ·

)and

ω1

(f (p+1), ·

). For this we use the following general estimate with optimal constants,

given in [5] or [6].Theorem E Let I be an arbitrary interval and let V ⊂ C(I) be a linear subspacesuch that ej ∈ V , for j = 0, 1, 2. If L : V → F(I) is a positive linear operator, thenwe have

|L(f, x)− f(x)| ≤ |f(x)| · |L(e0, x)− 1|+ |L(e1 − xe0, x)|h−1ω1(f, h)

+(16L(e0, x) +

1

2h2L((e1 − xe0)

2, x))hω1(f

′, h),(11)

for any f ∈ V ∩ C1(I), any x ∈ I and any 0 < h ≼ 12 length(I).

For the usual derivative we have

Theorem 4 For n ∈ N, p ∈ N, f ∈ Cp+1[0, 1], x ∈ [0, 1], 0 < h ≤ 12 , we have∣∣∣B(p)

n (f, x)− f (p)(x)∣∣∣ ≤ p(p− 1)

n|f (p)(x)|+ p

2n

|1− 2x|h

· ω1(f(p), h)

+

[1

6+

1

2h2

(n− p− p2

n2· x(1− x) +

3p2 + p

12n2

)]hω1(f

(p+1), h).(12)

Also, for n ∈ N, p ∈ N and f ∈ Cp+1[0, 1] we have

∥B(p)n (f)− f (p)∥ ≤ p(p− 1)

n∥f (p)∥+ p

2√n· ω1

(f (p),

1√n

)+

7

24· ω2

(f (p+1),

1√n

).(13)

Page 98: 2014 Volume 22 No. 1

98 R. Paltanea

Proof. Theorem 4 is a consequence of Theorem E applied to operator L = Ln,p andfunction f (p) by take into account (Bn)

(p)(f) = Ln,p(f(p)) and Corollaries 1, 2.

Theorem 5 For α ≥ 0, p = ⌈α⌉, f ∈ Cp[0, 1], n ∈ N, n ≥ 4 we have

∥DαBn(f)− Dαf∥ ≤ p(p− 1)

n∥f (p)∥+ p

2√n· ω1

(f (p),

1√n

)+

7

24ω1

(f (p+1),

1√n

).(14)

For α ∈ [0, 1) we obtain

∥DαBn(f)− Dαf∥ ≤ 1

2√nω1

(f ′,

1√n

)+

7

24ω1

(f ′′,

1√n

).(15)

Proof. The proof is based on relation (13) and is similar with the proof of Theorem3.

References

[1] G. Anastassiou, Fractional Korovkin theory, Chaos, Solitons, vol. 42, no. 4,2009, 2080-2094.

[2] G. Anastassiou, Fractional Voronovskaya type asymptotic expansions for quasi-interpolation neural network operators, CUBO A Mathematical Journal, vol.14, no. 3, 2011, 71-83.

[3] K. Diethelm, The Analysis of Fractional Differential Equations: AnApplication-Oriented Exposition Using Differential Operators of Caputo Type,Lecture Notes in Mathematics, Springer-Verlag, Berlin Heildeberg, 2010.

[4] B. Mond, Note: On the degree of approximation by linear positive operators, J.Approx. Theory, vol. 18, 1976, 304-306.

[5] R. Paltanea, Optimal estimates with moduli of continuity, Result. Math. vol.32, 1997, 318–331.

[6] R. Paltanea, Approximation Theory Using Positive Linear Operators,Birkhauser, Boston, 2004.

Radu PaltaneaTransilvania University of BrasovFaculty of Mathematics and Computer ScienceDepartment of Mathematics and Computer ScienceStr. Maniu Iuliu, nr. 50, Brasov - 500091, Romaniae-mail: [email protected]

Page 99: 2014 Volume 22 No. 1

General Mathematics Vol. 22, No. 1 (2014), 99–107

Uniform approximation of functions byBaskakov-Kantorovich operators 1

Gabriel Stan

Dedicated to the late Academician Professor Dr. Dimitrie D. Stancu

Abstract

In this article we give a generalization of the Baskakov- Durmeyer operatorusing Kantorovich method and we prove a theorem of uniform convergence ofthese operators.

2010 Mathematics Subject Classification: 41A36, 41A35.Key words and phrases: positive linear operator, Baskakov operators,modulus

of continuity.

1 Introduction

The Baskakov operators Vn : C2 ([0,∞)) → C ([0,∞)) are given by

(1) Vn (f) (x) =∞∑k=0

vn,k (x) f(kn

), x ∈ [0,∞) , n ∈ N ,

where

(2) vn,k (x) =

(n+ k − 1

k

)xk · (1 + x)−(n+k) .

These operators were introduced by Baskakov [1] in 1957 and intensely studied byMastroianni [4], Schurer [7], Totik [8]. and Della Vecchia [2].

In 1985 Sahai and Prasad [6] introduced modified Baskakov operators in thefollowing integral form

(3) V ∗∗n (f) (x) = (n− 1)

∞∑k=0

vn,k (x)∞∫0

vn,k (t) f (t) dt, x ∈ [0,∞) , n ∈ N ,

1Received 30 June, 2014Accepted for publication (in revised form) 17 August, 2014

99

Page 100: 2014 Volume 22 No. 1

100 G. Stan

so that the integral exists and the series is convergent.The purpose of this paper is to construct a Kantorovich type operator of higher

order and obtain a uniform convergence theorem for this operator. Definition andsome properties of these operators Kantorovich Baskakov are presented in Sections2 and 3, and the main results on the uniform convergence are given in Section 4.

Uniform convergence theorems for various operators were given by Holhos in [3]and Paltanea [5] .

2 Basic notations. Generalized Kantorovich operators

We consider a fixed N ∈ N, and define the function space

(4) CN :=f ∈ C ([0,∞)) : ∃ lim

x→∞f(x)1+xN finite

,

endowed with the norm

(5) ∥f∥∗N := supx≥0

|f(x)|1+xN .

Remark 1 We mention that (CN , ∥•∥∗N ) is a Banach space.

For r ∈ N we consider(i) Dr is the differentiation operator of r order,

Dr (f) (x) = f (r) (x) , f ∈ Cr ([0,∞)) , x ∈ [0,∞) ,

(ii) Ir is the antiderivative operator of r order,

Ir (f) (x) =

x∫0

(x− u)r−1

(r − 1)!f (u) du, f ∈ C ([0,∞)) , x ∈ [0,∞) .

Lemma 1 For r ∈ N we havei) (Dr Ir) (f) = f, for all f ∈ C ([0,∞)),ii) (Ir Dr) (f) = f, for all f ∈ Cr ([0,∞)), such that f (i) (0) = 0 for any

i = 0, 1, ....r − 1.

Proof. It is clear.Further note en (x) = xn, x ∈ [0,∞) , n ∈ N and (n)i = n·(n− 1)·...·(n− i+ 1) ,

i ∈ N.

Definition 1 Let r, n ∈ N. Define V rn : CN → C ([0,∞))

(6) V rn :=

(n−1)r+1

(n+r−1)r+1·Dr V ∗∗

n Ir, n, r ∈ N .

Theorem 1 Let r, n ∈ N. For f ∈ CN we have

(7) V rn (f) (x) = (n− r − 1)

∞∑k=0

vn+r,k (x)∞∫0

vn−r,k+r (t) f (t) dt, x ∈ [0,∞) .

Page 101: 2014 Volume 22 No. 1

Uniform approximation of functions by Baskakov-Kantorovich operators 101

Proof. Let f ∈ CN . We have

(V ∗∗n Ir) (f) (x) = (n− 1)

∞∑k=0

vn,k (x)

∞∫0

vn,k (t)

t∫0

(t− u)r−1

(r − 1)!f (u) dudt

= (n− 1)

∞∑k=0

vn,k (x)

∞∫0

f (u)

∞∫u

vn,k (t)(t− u)r−1

(r − 1)!dtdu.

In the last equality, we used the change of the domaint ∈ [0,∞)

u ∈ [0, t]in

u ∈ [0,∞)

t ∈ [u,∞)

By induction we prove with regard to s, 0 ≤ s ≤ r − 1 the following relation

(8)

Ds (V ∗∗n Ir) (f) (x)

=(n+ s− 1)s+1

(n− 1)s

∞∑k=0

vn+s,k (x)

∞∫0

f (u)

∞∫u

vn−s,k+s (t)(t− u)r−s−1

(r − s− 1)!dtdu.

For s = 0 this relation is clear. Supposing that it is true for s < r − 1 and prove itfor s+ 1. By a simple computation we obtain

D (vn,k) (x) = n [vn+1,k−1 (x)− vn+1,k (x)] ,

where we consider vn,k (x) = 0, for k < 0 and for any x ∈ [0,∞) . Also, for anyn, k, s ∈ N we have

lima→∞

[vn−k,k+s+1 (a)

(a− u)r−s−1

(r − s− 1)!

]= 0

Denote

Gsn,k (u) =

∞∫u

vn,k (t)(t− u)s

s!dt.

It follows(9)

Ds+1 (V ∗∗n Ir) (f) (x)

=(n+ s− 1)s+1

(n− 1)s

∞∑k=0

D (vn+s,k) (x)

∞∫0

f (u)Gr−s−1n−s,k+s (u) du

=(n+s−1)s+1

(n− 1)s

∞∑k=0

(n+s) [vn+s+1,k−1 (x)−vn+s+1,k (x)]

∞∫0

f (u)Gr−s−1n−s,k+s (u) du

=(n+ s)s+2

(n− 1)s

∞∑k=0

vn+s+1,k (x)

∞∫0

f (u)[Gr−s−1

n−s,k+s+1 (u)−Gr−s−1n−s,k+s (u)

]du.

Page 102: 2014 Volume 22 No. 1

102 G. Stan

We have

Gr−s−1n−s,k+s+1 (u)−Gr−s−1

n−s,k+s (u)

=

∞∫u

[vn−s,k+s+1 (t)− vn−s,k+s (t)](t− u)r−s−1

(r − s− 1)!dt

=

∞∫u

− 1

n− s− 1D (vn−s−1,k+s+1) (t)

(t− u)r−s−1

(r − s− 1)!dt

= − 1

n− s− 1lima→∞

[vn−s−1,k+s+1 (a)

(a− u)r−s−1

(r − s− 1)!

]

+1

n− s− 1

∞∫u

vn−s−1,k+s+1 (t)(t− u)r−s−2

(r − s− 2)!dt

=1

n− s− 1Gr−s−2

n−s−1,k+s+1 (u) .

Replacing in (9) we obtain

Ds+1 (V ∗∗n Ir) (f) (x)

=(n+ s)s+2

(n− 1)s+1

∞∑k=0

vn+s+1,k (x)

∞∫0

f (u)Gr−s−2n−s−1,k+s+1 (u) dtdu

which is the analogous of relation (8) for s+1 instead of s. For s = r− 1 we obtain

Dr−1 (V ∗∗n Ir) (f) (x)

=(n+ r − 2)r(n− 1)r−1

∞∑k=0

vn+r−1,k (x)

∞∫0

f (u)

∞∫u

vn−r+1,k+r−1 (t) dtdu.

Hence

Dr (V ∗∗n Ir) (f) (x) =

(n+ r − 2)r(n− 1)r−1

∞∑k=0

(n+ r − 1) [vn+r,k−1 (x)− vn+r,k (x)]

·∞∫0

f (u)

∞∫u

vn−r+1,k+r−1 (t) dtdu

=(n+ r − 1)r+1

(n− 1)r−1

∞∑k=0

vn+r,k (x)

·∞∫0

f (u)

∞∫u

[vn−r+1,k+r (t)− vn−r+1,k+r−1 (t)] dtdu

=(n+ r − 1)r+1

(n− 1)r

∞∑k=0

vn+r,k (x)

∞∫0

f (u) vn−r,k+r (u) du.

Page 103: 2014 Volume 22 No. 1

Uniform approximation of functions by Baskakov-Kantorovich operators 103

From (6) we obtain (7).

Remark 2 This operator is linear and positive.

Remark 3 For f ∈ CN the integral exists and the series is convergent.

3 Moments and their recursion

Lemma 2 For any n, r ∈ N, n sufficiently high and x ∈ [0,∞) we havei) V r

n (e0) (x) = 1.ii) V r

n (e1) (x) = x+ r+1n−r−2 (2x+ 1) .

Proof. It is clear.

Lemma 3 For any n, r, s ∈ N, n sufficiently high and x ∈ [0,∞) we have(10)

V rn (es+1) (x)=

1

n−r−s−2x (x+1) ·D (V r

n (es)) (x)+[(n+r)x+r+s+1]V rn (es) (x) .

Proof. From (7) we have

V rn (es) (x) = (n− r − 1)

∞∑k=0

vn+r,k (x)

∞∫0

vn−r,k+r (t) · tsdt

(11)

= (n− r − 1)∞∑k=0

vn+r,k (x)

∞∫0

(n+ k − 1

k + r

)tk+r+s · (1 + t)−(n+k) dt

= (n−r−1)

∞∑k=0

vn+r,k (x)(n+k−1)!

(k+r)! (n−r−1)!· (k+r+s)! (n−r−s−2)!

(n+k−1)!

=

∞∑k=0

vn+r,k (x) ·(k + r + s)s(n− r − 2)s

= (1 + x)−(n+r)∞∑k=0

(n+ r + k − 1

k

)(x

1 + x

)k

·(k + r + s)s(n− r − 2)s

.

and

(12) V rn (es+1) (x) = (1 + x)−(n+r)

∞∑k=0

(n+r+k−1

k

) (x

1+x

)k· (k+r+s+1)s+1

(n−r−2)s+1.

We haveD (V r

n (es)) (x) = T1 + T2

where

T1 = −n+ r

1 + xV rn (es) (x)

Page 104: 2014 Volume 22 No. 1

104 G. Stan

and

T2 = (1 + x)−(n+r)∞∑k=0

(n+ r + k − 1

k

)· k ·

(x

1 + x

)k−1

· 1

(1 + x)2·(k + r + s)s(n− r − 2)s

= (1 + x)−(n+r)∞∑k=0

(n+ r + k − 1

k

)·(

x

1 + x

)k

· 1

x (1 + x)

·[(k + r + s+ 1)s+1

(n− r − 2)s− (r + s+ 1)

(k + r + s)s(n− r − 2)s

]=

n− r − s− 2

x (1 + x)· V r

n (es+1) (x)−r + s+ 1

x (1 + x)V rn (es) (x) .

Hence

D (V rn (es)) (x) =

n− r − s− 2

x (1 + x)· V r

n (es+1) (x)−(n+ r)x+ r + s+ 1

x (1 + x)V rn (es) (x) ,

which implies (10).

Corollary 1 For any n, r, s ∈ N, n sufficiently high and x ∈ [0,∞) we have

(13) V rn (es) (x) = xs +Rn,r,s (x) ,

where

|Rn,r,s (x)| ≤ (1 + xs)M

n,

and M is a constant which depends on s and r.

Lemma 4 For any n, r, s ∈ N, n sufficiently high and x ∈ [0,∞) we have

(14) V rn (|ts − xs|) (x) ≤

√(1+x2s)·M

n , ,

where M is a constant which depends on s and r.

Proof. Using Corollary (1) we have

V rn (|ts − xs|) (x) ≤

√V rn

((es − xs)2

)(x) · V r

n (e0) (x)

≤√V rn (e2s) (x)− 2xsV r

n (es) (x) + x2s

≤√

(1 + x2s)M

n.

Lemma 5 For any n, r ∈ N, n sufficiently high and x ∈ [0,∞) we have

V rn

(1

1 + t

)(x) ≤ 1

1 + x· n+ r

n+ r − 1.

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Uniform approximation of functions by Baskakov-Kantorovich operators 105

Proof. Using Theorem (1) we have

V rn

(1

1 + t

)(x) = (n− r − 1)

∞∑k=0

vn+r,k (x)

∞∫0

vn−r,k+r (t) ·1

1 + tdt

= (n− r − 1)∞∑k=0

vn+r,k (x) ·1

n+ k

=1

1 + x

∞∑k=0

vn+r−1,k (x) ·(n− r − 1) (n+ r + k − 1)

(n+ k) (n+ r − 1)

≤ 1

1 + x· n+ r

n+ r − 1.

Lemma 6 For any n, r ∈ N, n sufficiently high and x ∈ [0,∞) we have

V rn (|ln (t+ 1)− ln (x+ 1)|) (x) ≤

√2r + 3√

n+ r − 2.

Proof. From the geometric-logaritmic-arithmetic mean inequality we have

|ln (t+ 1)− ln (x+ 1)| ≤ |t− x|√(1 + t) (1 + x)

=

∣∣∣∣∣√

1 + x

1 + t−√

1 + t

1 + x

∣∣∣∣∣ .Whence, using Lemma (5) we have

V rn (|ln (t+ 1)− ln (x+ 1)|) (x)

≤√V rn

([|ln (t+ 1)− ln (x+ 1)|]2

)(x)

√V rn

(1 + x

1 + t

)(x)− 2 + V r

n

(1 + t

1 + x

)(x)

√(1 + x) · 1

1 + x· n+ r

n+ r − 1− 2 +

1

1 + x

(1 + x+

r + 1

n− r − 2(2x+ 1)

)≤

√2r + 3√

n+ r − 2.

4 Main results

Theorem 2 For any r ∈ N the Baskakov-Kantorovich operators V rn : CN → CN

have the property that if f (ex − 1) · 11+(ex−1)N

is uniformly continuous on [0,∞) then

∥V rn (f)− f∥∗N → 0 when n→ ∞.

Moreover, for every f ∈ CN , r ∈ N and n sufficiently high one has

∥V rn (f)− f∥∗N ≤ ∥f∥∗N · C√

n+ 2ω

(f (ex − 1) · 1

1 + (ex − 1)N,

√2r + 3√

n+ r − 2

),

Page 106: 2014 Volume 22 No. 1

106 G. Stan

where C ∈ R is a constant depending on N and r.

Proof. Let f (x) = 11+(ex−1)N

· f (ex − 1) and g (x) = ln (x+ 1) .We notice that(f g

)(x) = f (x) · 1

1+xN .

We consider the usual first order continuity modulus of the function f given by

ω(f , δ)= sup

∣∣∣f (x)− f (y)∣∣∣ : x, y ∈ [0,∞) , |x− y|

.

Taking into account the properties of this modulus and the uniform continuity of fwe have∣∣∣∣ 1

1 + tNf (t) − 1

1 + xNf (x)

∣∣∣∣= ∣∣∣f (g (t))−f (g (x))∣∣∣ ≤ (1+ |g (t)−g (x)|δ

)ω(f , δ).

Further we obtain

|f (t)− f (x)| =

∣∣∣∣f (t)− 1 + xN

1 + tNf (t) +

1 + xN

1 + tNf (t)− f (x)

∣∣∣∣≤ 1

1+tN· |f (t)| ·

∣∣tN−xN∣∣+(1+xN) · ∣∣∣∣ 1

1+tNf (t)− 1

1+xNf (x)

∣∣∣∣≤ ∥f∥∗N ·

∣∣tN − xN∣∣+ (1 + xN

)·(1 +

|g (t)− g (x)|δ

)ω(f , δ).

Applying the operators V rn and using the Lemmas (4) and (6) we obtain

|V rn (f) (x)− f (x)|

≤ V rn (|f (t)− f (x)|) (x)

≤ ∥f∥∗N V rn

(∣∣tN − xN∣∣) (x) + (1 + xN

)(1 +

1

δV rn (|g (t)− g (x)|) (x)

)ω(f , δ)

≤ ∥f∥∗N ·√

(1 + x2N ) ·Mn

+(1 + xN

)·(1 +

1

δ·

√2r + 3√

n+ r − 2

)ω(f , δ).

Whence we obtain

1

1 + xN· |V r

n (f) (x)− f (x)| ≤ ∥f∥∗N ·√M

n+

(1 +

1

δ·

√2r + 3√

n+ r − 2

)ω(f , δ).

Choosing δ =√2r+3√n+r−2

the Theorem is proved.

Theorem 3 For any r ∈ N the Baskakov integral operators V ∗∗n : CN → CN have

the property that if Dr (f) (ex − 1) · 11+(ex−1)N

is uniformly continuous on [0,∞) then

∥Dr (V ∗∗n ) (f)−Dr (f)∥∗N → 0 when n→ ∞.

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Uniform approximation of functions by Baskakov-Kantorovich operators 107

Moreover, for every f ∈ CN , r ∈ N and n sufficiently high one has

∥Dr (V ∗∗n ) (f)−Dr (f)∥∗N

≤ ∥Dr (f)∥∗N · C√n+ 2ω

(Dr (f) (ex − 1) · 1

1 + (ex − 1)N,

√2r + 3√

n+ r − 2

),

where C ∈ R is a constant depending on N and r.

Proof. It obviously follows from the previous Theorem (2).

References

[1] Baskakov, V.A., An example of a sequence of linear positive operators in thespace of continuous functions, Dokl. Acad. Nauk. SSSR, 113 (1957), 249-251.

[2] Della Vecchia, B., On the monotonicity of the derivates of the sequences ofFavard and Baskakov operators, Ricerche di Matematica, 36 (1987), 263-269.

[3] Holhos, A., Uniform weighted approximation by positive linear operators, Stud.Univ. Babes-Bolyai Math., 56 (3) (2011), 135-146.

[4] Mastroianni, G., Su una classe di operatori lineari e pozitivi, Rend. Accad.Scien., Fis. Mat., Napoli, Serie IV, 48 (1980), Anno CXX, 217-235.

[5] Paltanea, R., Estimates for general positive linear operators on non-compactinterval using weighted moduli of continuity, Stud. Univ. Babes-Bolyai Math.,56 (3) (2011), 497-504.

[6] Sahai, A., Prasad, G., On simultaneous approximation by modified Lupas oper-ators, J. Approx. Theory, 45 (1985), 122-128.

[7] Schurer, F., Linear positive operators in approximation theory, Math. Inst.Techn. Univ. Delft. Report., 1962.

[8] Totik, V. Uniform approximation by Szasz-Mirakjan type operators, Acta Math.Hungar., 41 (1983),no. 3-4, 291-307.

Gabriel StanTransilvania University of BrasovFaculty of Mathematics and InformaticsDepartment of Mathematics50 Maniu Iuliu, RO-500091 Brasov, Romaniae-mail: [email protected]

Page 108: 2014 Volume 22 No. 1
Page 109: 2014 Volume 22 No. 1

General Mathematics Vol. 22, No. 1 (2014), 109–118

Properties of the intermediate point from a mean-valuetheorem 1

Dorel I. Duca, Emilia-Loredana Pop

Dedicated to the late Academician Professor Dr. Dimitrie D. Stancu

Abstract

In this paper we consider a mean-value theorem. For this one we studythe properties of the intermediate point and formulate some important resultsrelated to its derivatives.

2010 Mathematics Subject Classification: 26A24.

Key words and phrases: mean-value theorem, intermediate point,differentiability.

1 Introduction and Preliminaries

In mathematical analysis for the intermediate point from the mean-value theoremswas given a special interest after these theorems were proved. We remember someauthors in this field, like E. Abason [1, 2, 3], U. Abel and M. Ivan [4, 5], A. M.Acu [6, 7], D. Acu and P. Dicu [8], D.I. Duca [10, 11, 12], D.I. Duca and E. Duca[13], D.I. Duca and O. Pop [14, 15, 16], Al. Lupas [17], D. Pompeiu [18], E.C. Popa[19, 20], T. Tchakaloff [21, 22], L. Teodoriu [23, 24], T. Trif [25].

In this paper we study the properties related to the intermediate point from amean-value theorem less known. This theorem follows.

Theorem 1 Let a, b ∈ R with a < b and f : [a, b] → R be a continuous function on[a, b] and derivable on ]a, b[. Then there exists a point c ∈]a, b[ such that

(1) f(a)− f(c) = f ′(c)(c− b).

1Received 14 July, 2014Accepted for publication (in revised form) 18 August, 2014

109

Page 110: 2014 Volume 22 No. 1

110 D.I. Duca, E.-L. Pop

Proof. Let us consider the function g : [a, b] → R defined by

g(x) = (x− b)(f(x)− f(a)), ∀x ∈ [a, b].

From the hypothesis of the theorem follows that the function g is continuous on [a, b]and derivable on ]a, b[. Also, one has that g(a) = g(b) = 0. Consequently, we canapply Rolle’s theorem and follows that there exists at least one point c ∈]a, b[ suchthat g′(c) = 0. This means f(c)− f(a) + (c− b)f ′(c) = 0 and consequently

f(a)− f(c) = f ′(c)(c− b).

2 Main results

Let I ⊆ R be an interval, a ∈ I and f : I → R a differentiable function on I. Thenfor each x ∈ I \ a, there exists cx ∈ I such that

f(a)− f(cx) = f ′(cx)(cx − x).

If, for each x ∈ I \a, we choose one cx from the interval with the extremities xand a which satisfies this relation, then we can define the function c : I\a → I\aby

c(x) = cx, for all x ∈ I \ a.

This function c satisfies relation

(2) f(a)− f(c(x)) = f ′(c(x))(c(x)− x), for all x ∈ I \ a.

If x ∈ I \ a tends to a, because

|c(x)− a| ≤ |x− a|,

we havelimx→a

c(x) = a.

Then the function c : I → I defined by

(3) c(x) =

c(x), if x ∈ I \ aa, if x = a

is continuous at x = a.One of the purpose of this paper is to establish under which circumstances the

function c is differentiable at the point x = a and to compute its derivative c′(a). Ifthere exist several functions c which satisfies (2), does the derivative of the functionc at x = a depend upon the function c we choose?

Since for x ∈ I \ ac(x)− c(a)

x− a=c(x)− a

x− a

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Properties of the intermediate point from a mean-value theorem 111

if we denote by

θ(x) =c(x)− a

x− a,

thenθ(x) ∈]0, 1[, c(x) = a+ (x− a)θ(x)

and hence

(4) f(a+ (x− a)θ(x))− f(a) = (x− a)[1− θ(x)]f ′(a+ (x− a)θ(x)).

Obviously, the function c : I → I defined by (3) is differentiable at x = a if andonly if the function θ : I \ a →]0, 1[ defined by

θ(x) =c(x)− c(a)

x− a=c(x)− a

x− a, for all x ∈ I \ a

has limit at the point x = a. Moreover, if the function c is differentiable at x = a,then

c′(a) = limx→a

θ(x).

In what follows, we assume that f is two times differentiable on I, the functionf ′′ is continuous on intI and f ′(a) = 0. In this hypotheses, passing to limit withx→ a in relation (2) written in the form

f(a+ (x− a)θ(x))− f(a)

x− a= [1− θ(x)]f ′(a+ (x− a)θ(x)),

for all x ∈ I \ a, we obtain

f ′(a) limx→a

θ(x) = (1− limx→a

θ(x))f ′(a),

hence

limx→a

θ(x) =1

2.

Let’s consider the function θ : I →]0, 1[ defined by

θ(x) =

θ(x), if x = a12 , if x = a.

Obviously, the function θ is continuous at a. Next, we study the problem ofthe differentiability of the function θ at x = a. For this we remark that, by usingTaylor’s formula, we have that

(5) f(a+(x−a)θ(x)) = f(a)+f ′(a)

1!(x−a)θ(x)+ f ′′(a+ (x− a)θ(x)θ)

2!(x−a)2θ2(x)

and

(6) f ′(a+ (x− a)θ(x)) = f ′(a) +f ′′(a+ (x− a)θ(x)θ)

1!(x− a)θ(x),

Page 112: 2014 Volume 22 No. 1

112 D.I. Duca, E.-L. Pop

where θ, θ ∈]0, 1[.Taking into account the relations (5) and (6), the relation (4) becomes

f ′(a)

1!θ(x) +

f ′′(a+ (x− a)θ(x)θ)

2!(x− a)θ2(x)(7)

= [1− θ(x)][f ′(a) +f ′′(a+ (x− a)θ(x)θ)

1!(x− a)θ(x)],

or equivalently,

2f ′(a)θ(x)− 1

2

x− a=−f

′′(a+ (x− a)θ(x)θ)

2!θ2(x)+

f ′′(a+ (x− a)θ(x)θ)

1!θ(x)[1− θ(x)].

But θ(a) =1

2and so, for every x ∈ I \ a, we have

θ(x)−θ(a)x− a

=1

2f ′(a)

−f

′′(a+(x−a)θ(x)θ)2!

θ2(x)+

f ′′(a+(x−a)θ(x)θ)1!

θ(x)[1−θ(x)]

.

Here we pass to limit with x→ a and obtain

limx→a

θ(x)− θ(a)

x− a= − 1

2f ′(a)limx→a

f ′′(a+ (x− a)θ(x)θ)

2!θ2(x)

+1

2f ′(a)limx→a

f ′′(a+ (x− a)θ(x)θ)

1!θ(x)[1− θ(x)]

which means

θ′(a) = − 1

2f ′(a)

f ′′(a)

2θ2(a) +

1

2f ′(a)f ′′(a)θ(a)[1− θ(a)],

hence

θ′(a) =

f ′′(a)

16f ′(a).

Consequently, we have the following theorem.

Theorem 2 Let I ⊆ R be an interval, a an interior point of I and f : I → R afunction satisfying the conditions

(a) the function f is two times differentiable on I,(b) the function f ′′ is continuous on intI,(c) f ′(a) = 0.

Then10 There exists a function c : I \ a → I \ a such that (2) is true.20 There exists a function θ : I \ a →]0, 1[ such that (4) is true.30 The function θ : I → [0, 1] defined by

θ(x) =

θ(x), if x ∈ I \ a12 , if x = a

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Properties of the intermediate point from a mean-value theorem 113

is differentiable at x = a and

θ′(a) =

f ′′(a)

16f ′(a).

40 The function c : I → I defined by

c(x) =

c(x), if x ∈ I \ aa, if x = a

is two times differentiable at x = a and

c′(a) =1

2and c′′(a) =

f ′′(a)

8f ′(a).

For studying the differentiability of order two of the function θ at x = a weassume that f is three times differentiable on I, the function f ′′′ is continuous onintI and f ′(a) = 0. Let’s consider the following Taylor’s formula

f(a+ (x− a)θ(x)) = f(a) +f ′(a)

1!(x− a)θ(x) +

f ′′(a)

2!(x− a)2θ2(x)

+f ′′′(a+ (x− a)θ(x)θ)

3!(x− a)3θ3(x),

and

f ′(a+ (x− a)θ(x)) = f ′(a) + f ′′(a)(x− a)θ(x)

+f ′′′(a+ (x− a)θ(x)θ)

2!(x− a)2θ2(x),

where θ, θ ∈]0, 1[. Then relation (4) becomes

f ′(a)

1!θ(x) +

f ′′(a)

2!(x− a)θ2(x) +

f ′′′(a+ (x− a)θ(x)θ)

3!(x− a)2θ3(x)

= [1− θ(x)][f ′(a) + f ′′(a)(x− a)θ(x) +f ′′′(a+ (x− a)θ(x)θ)

2!(x− a)2θ2(x)]

or equivalently,

f ′(a)[2θ(x)−1]=θ(x)

[1− 3

2θ(x)

]f ′′(a)(x−a)+ 1

2θ2(x)

×

−f

′′′(a+(x−a)θ(x)θ)3

θ(x)+[1−θ(x)]f ′′′(a+(x− a)θ(x)θ)

(x−a)2

where we divide by x− a and by the fact that θ(a) = 12 we obtain

2f ′(a)θ(x)−θ(a)x−a

=θ(x)

[1− 3

2θ(x)

]f ′′(a)+

1

2θ2(x)

×

−f

′′′(a+(x−a)θ(x)θ)3

θ(x)+[1−θ(x)]f ′′′(a+(x−a)θ(x)θ)

(x−a).

(8)

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114 D.I. Duca, E.-L. Pop

Passing to limit with x→ a in the previous relation we get

2f ′(a)θ′(a) = θ(a)

[1− 3

2θ(a)

]f ′′(a).

Consequently

θ′(a) =

f ′′(a)

16f ′(a).

Now, the relation (8) can be rewritten as follows

θ(x)−θ(a)x−a − f ′′(a)

16f ′(a)

x− a= −3

4

f ′′(a)

f ′(a)

θ(x)− 12

x− a

[θ(x)− 1

6

]+θ2(x)

4f ′(a)

f ′′′(a+ (x− a)θ(x)θ)[1− θ(x)]− f ′′′(a+ (x− a)θ(x)θ)

3θ(x)

,

for all x ∈ I \ a.If, in this relation, we pass to limit with x→ a we obtain

1

2θ′′(a) = −3

4

f ′′(a)

f ′(a)θ′(a)

[θ(a)− 1

6

]+

θ2(a)

4f ′(a)

f ′′′(a)[1− θ(a)]− f ′′′(a)

3θ(a)

.

Using that θ(a) =1

2and θ

′(a) =

f ′′(a)

16f ′(a)we obtain

θ′′(a) =

4f ′(a)f ′′′(a)− 3 [f ′′(a)]2

96 [f ′(a)]2.

Hence, we have the following result.

Theorem 3 Let I ⊆ R be an interval, a an interior point of I and f : I → R afunction satisfying the conditions

(a) the function f is three times differentiable on I,

(b) the function f ′′′ is continuous on intI,

(c) f ′(a) = 0.

Then the function θ is two times differentiable at x = a,

θ′(a) =

f ′′(a)

16f ′(a)

and

θ′′(a) =

4f ′(a)f ′′′(a)− 3 [f ′′(a)]2

96 [f ′(a)]2.

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Properties of the intermediate point from a mean-value theorem 115

For studying the differentiability of order three of the function θ at x = a weassume that f is four times differentiable, f (4) is continuous on intI and f ′(a) = 0.Let’s consider the following Taylor’s formula

f(a+ (x− a)θ(x)) = f(a) +f ′(a)

1!(x− a)θ(x) +

f ′′(a)

2!(x− a)2θ2(x)

+f ′′′(a)

3!(x− a)3θ3(x) +

f (4)(a+ (x− a)θ(x)θ)

4!(x− a)4θ4(x),

and

f ′(a+ (x− a)θ(x)) = f ′(a) + f ′′(a)(x− a)θ(x) +f ′′′(a)

2!(x− a)2θ2(x)

+f (4)(a+ (x− a)θ(x)θ)

3!(x− a)3θ3(x),

where θ, θ ∈]0, 1[. Then relation (4) becomes, after the calculus,

2f ′(a)θ(x)− θ(a)

x− a= θ(x)

[1− 3

2θ(x)

]f ′′(a)

(9)

+θ2(x)

2

[1− 4

3θ(x)

]f ′′′(a)(x− a)

+θ3(x)

6

−f

(4)(A(x))

4θ(x) + [1− θ(x)]f (4)(B(x))

(x− a)2,

where A(x) = a+ (x− a)θ(x)θ and B(x) = a+ (x− a)θ(x)θ.Passing to limit with x→ a in the previous relation we get

θ′(a) =

f ′′(a)

16f ′(a).

Now, the relation (9) can be rewritten as follows

θ(x)−θ(a)x−a − f ′′(a)

16f ′(a)

x−a=−3

4

f ′′(a)

f ′(a)

θ(x)− 12

x−a

[θ(x)− 1

6

]+θ2(x)

4

[1− 4

3θ(x)

]f ′′′(a)

f ′(a)(10)

+θ3(x)

12

−f

(4)(A(x))

4θ(x)+[1−θ(x)]f (4)(B(x))

(x−a)f ′(a)

.

If, in this relation, we pass to limit with x→ a we obtain

θ′(a) =

f ′′(a)

16f ′(a)

and

θ′′(a) =

4f ′(a)f ′′′(a)− 3 [f ′′(a)]2

96 [f ′(a)]2.

Page 116: 2014 Volume 22 No. 1

116 D.I. Duca, E.-L. Pop

Then relation (10) can be rewritten as follows

θ(x)−θ(a)x−a − f ′′(a)

16f ′(a)

x− a− 4f ′(a)f ′′′(a)− 3 [f ′′(a)]2

96 [f ′(a)]2

x− a(11)

=[−36θ

2(x) + 24θ(x)− 3]f ′′(a) + [12θ

2(x)− 16θ

3(x)]f ′′′(a)(x− a)

48f ′(a)(x− a)2

− 4f ′(a)f ′′′(a)− 3 [f ′′(a)]2

96 [f ′(a)]2 (x− a)

+θ3(x)

12f ′(a)

−f

(4)(A(x))

4θ(x) + [1− θ(x)]f (4)(B(x))

.

Passing to limit with x→ a we get

1

3!θ′′′(a)= lim

x→a

[−72θ(x)θ′(x)+24θ

′(x)]f ′′(a)+[12θ

2(x)−16θ

3(x)]f ′′′(a)

96f ′(a)(x−a)(12)

+ limx→a

[24θ′(x)−48θ

2(x)θ

′(x)]f ′′′(a)(x−a)−48θ

′′(a)f ′(a)

96f ′(a)(x−a)− f (4)(a)

256f ′(a)

and so

θ′′′(a) = − 3

128

[f ′′(a)]3 + [f ′(a)]2 f (4)(a)

[f ′(a)]3− 3

4

f ′′(a)

f ′(a)θ′′(a).

Hence, we have the following result.

Theorem 4 Let I ⊆ R be an interval, a an interior point of I and f : I → R afunction satisfying the conditions

(a) the function f is four times differentiable on I,(b) the function f (4) is continuous on intI,(c) f ′(a) = 0.

Then the function θ is three times differentiable at x = a,

θ′(a) =

f ′′(a)

16f ′(a), θ

′′(a) =

4f ′(a)f ′′′(a)− 3 [f ′′(a)]2

96 [f ′(a)]2

and

θ′′′(a) = − 3

128

[f ′′(a)]3 + [f ′(a)]2 f (4)(a)

[f ′(a)]3− 3

4

f ′′(a)

f ′(a)θ′′(a).

3 Conclusions and further challenges

For the function intermediate point given by formula (2) we gave sufficient differ-entiability conditions of superior order. The differentiability of order n ≥ 4 needssome complicated calculus that will be given in a following paper.

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Properties of the intermediate point from a mean-value theorem 117

References

[1] E. Abason, Sur le theoreme des accroissements finis, Bulletin de Mathematiqueset de Physique Pures et Appliquees de l’Ecole Polytechnique de Bucharest, vol.1, no. 1, 1929, 4-10.

[2] E. Abason, Sur le theoreme des accroissements finis, Bulletin de Mathematiqueset de Physique Pures et Appliquees de l’Ecole Polytechnique de Bucharest, vol.1, no. 2, 1930, 81-86.

[3] E. Abason, Sur le theoreme des accroissements finis, Bulletin de Mathematiqueset de Physique Pures et Appliquees de l’Ecole Polytechnique de Bucharest, vol.1, no. 3, 1930, 149-152.

[4] U. Abel, M. Ivan, Asymptotic Expansion of a Sequence of Divided Differenceswith Application to Positive Linear Operators, Journal of Computational Anal-ysis and Applications, vol. 7, no. 1, 2005, 89-101.

[5] U. Abel, M. Ivan, The Differential Mean Value of Divided Differences, Journalof Mathematical Analysis and Applications, vol. 325, 2007, 560-570.

[6] A. M. Acu, About an Intermediate Point Property in Some Quadrature Formu-las, Acta Universitatis Apulensis, no. 15, 2008, 19-32.

[7] A. M. Acu, An Intermediate Point Property in the Quadrature Formulas, Pro-ceedings of International Workshop New Trends in Approximation, Optimiza-tion and Classification, Sibiu, October 1-5, 2008, 9-20.

[8] D. Acu, P. Dicu, About some Properties of Intermediate Point in certain Mean-Value Formulas, General Mathematics, vol. 10, no. 1-2, 2002, 51-64.

[9] D.I. Duca, Analiza Matematica (vol. I), Casa Cartii de Stiinta, Cluj-Napoca,2013.

[10] D.I. Duca, A Note on the Mean Value Theorem, Didactica Matematica, vol. 19,2003, 91-102.

[11] D. I. Duca, Properties of the Intermediate Point from the Taylor’s Theorem,Mathematical Inequalities & Applications, vol. 12, no. 4, 2009, 763-771.

[12] D.I. Duca, Proprietati ale punctului intermediar din teorema de medie a luiLagrange, Gazeta Matematica, seria A, XXVII(CVI) no. 4, 2009, 284-294.

[13] D.I. Duca, E. Duca, Exercitii si probleme de analiza matematica (vol. I), CasaCartii de Stiinta, 2007.

[14] D.I. Duca, O. Pop, Asupra punctului intermediar din teorema cresterilor finite,Lucrarile Seminarului de Didactica Matematicii, vol. 19, 2003, 91-102.

Page 118: 2014 Volume 22 No. 1

118 D.I. Duca, E.-L. Pop

[15] D.I. Duca, O. Pop, On the Intermediate Point in Cauchy’s Mean-Value Theo-rem, Mathematical Inequalities & Applications, vol. 9, no. 3, 2006, 375-389.

[16] D.I. Duca, O. Pop, Concerning the Intermediate Point in the Mean-Value The-orem, Mathematical Inequalities & Applications, vol. 12, no. 3, 2009, 499-512.

[17] Al. Lupas, Asupra teoremei cresterilor finite, Revista de matematica a elevilordin Timisoara, vol. 14, no. 2, 1975, 6-13.

[18] D. Pompeiu, Sur la theoreme des accroissements finis, Annales Scientiques del’Universite de Jassy, vol. 15, no. 3-4, 1928, 335-337.

[19] E.C. Popa, On a Mean Value Theorem (in romanian), MGB, vol. 4, 1989, 113-114.

[20] E.C. Popa, On a Property of Intermediary Point ”c” for a Mean Value Theorem(in romanian), MGB, vol. 8, 1994, 348-350.

[21] L. Tchakaloff, Sur la structure des ensemble lineaires definis par un certainepro- priete minimale, Acta Mathematica, vol. 63, 1934, 77-97.

[22] L. Tchakaloff, Uber den Rolleschen Satz angewandt auf lineare Kombinatio-nen, endlich viller Funktionen, C.R. du Premier Congres des mathematicienshongrois Budapest, 27.08-2.09.1950, Budapest, 1952, 591-594.

[23] L. Teodoriu, Sur une equation aux derivees partielle qui s’introduit dans unprobleme de moyenne, Comptes Rendus de l’Academie des Sciences, Paris, vol.191, 1930, 431-433.

[24] L. Teodoriu, Cercetari asupra teoremei cresterilor finite, Facultatea de Stiintedin Bucuresti, Teza de doctorat, nr. 71, Bucuresti, 1931.

[25] T. Trif, Asymptotic Behavior of Intermediate Points in certain Mean ValueTheorems, Journal of Mathematical Inequalities, vol. 2, no. 2, 2008, 151-161.

Dorel I. DucaFaculty of Mathematics and Computer ScienceBabes-Bolyai University1 Mihail Kogalniceanu street, 400084 Cluj-Napoca, [email protected]

Emilia-Loredana PopTheological Advent HighSchool ”Maranatha”Micro II/2 Campului street, 400664, Cluj-Napoca, Romaniapop emilia [email protected]

Page 119: 2014 Volume 22 No. 1

General Mathematics Vol. 22, No. 1 (2014), 119–123

Characterization theorems of Jacobi and Laguerrepolynomials 1

Ioan Tincu

Dedicated to the late Academician Professor Dr. Dimitrie D. Stancu

Abstract

In this paper we prove a property of the Jacobi polynomials using theinterpolation polynomial of Hermite and prove a property of the Laguerrepolynomials.

2010 Mathematics Subject Classification: 33C45, 33C52.Key words and phrases: Jacobi, Laguerre, Hermite interpolation formula.

1 Introduction

For α > −1, let R(α)n (x) = 2F1

(−n, n+ 2α+ 1;α+ 1 =

1− x

2

), x ∈ [−1, 1], be the

Jacobi polynomials of degree n normalized by R(α)n (1) = 1 and

L(α)n (x) =

Γ(α+ 1)

Γ(n+ α+ 1)exx−α(e−xxn+α)(n), x ≥ 0 be the polynomials Laguerre of

degree n normalized L(α)n (0) = 1.

The following formulas are known:

(1− x2)y′′(x)− (2α+ 2)xy′(x) + n(n+ 2α+ 1)y(x) = 0, y(x) = R(α)

n (x),(1)

R(α)n+1(x) =

2n+ 2α+ 1

n+ 2α+ 1xR(α)

n (x)− n

n+ 2α+ 1R

(α)n−1(x),(2)

(1− x2)d

dxR(α)

n (x) = −nxR(α)n (x) + nR

(α)n−1(x),(3)

xy′′(x) + (1 + α− x)y′(x) + ny(x) = 0, y(x) = L(α)n (x),(4)

(n+ α+ 1)L(α)n+1(x) + (x− α− 2n− 1)L(α)

n (x) + nL(α)n−1(x) = 0,(5)

xd

dxL(α)n (x) = nL(α)

n (x)− nL(α)n−1(x)(6)

1Received 3 July, 2014Accepted for publication (in revised form) 6 September, 2014

119

Page 120: 2014 Volume 22 No. 1

120 I. Tincu

and the Christoffel-Darboux formula’s

(7)

n∑k=0

wkR(α)k (x)R

(α)k (y) = λn

R(α)n (y)R

(α)n+1(x)−R

(α)n (x)R

(α)n+1(y)

x− y, x = y

where1

wk=

∫ 1

−1[R

(α)k (x)]2(1− x2)αdx, k = 0, n,

λn =Γ(n+ 2α+ 2)

22α+1Γ2(α+ 1)n!

From (7), for y → x we obtain

(8)n∑

k=0

wk[R(α)k (x)]2 = λn[R

(α)n (x)[R

(α)n+1(x)]

′ −R(α)n+1(x)[R

(α)n (x)]′].

2 Main Results

Let f : [−1, 1] → R, f(x) = (1− x2)Rαn(x)[R

(α)n+1(x)]

′ −R(α)n+1(x)[R

(α)n (x)]′.

Using (2), (3) and (8) one finds

(9) f(x) = (n+ 1)[R(α)n (x)]2 − xR(α)

n (x)R(α)n+1(x)− nR

(α)n−1(x)R

(α)n+1(x), f ∈ Π2n+2.

Notation: R(α)n (x) = Rn(x), n ∈ N.

According to Hermite interpolation formula

f(x) = H2n+2(x1, x1, x2, x2, ..., xn+1, xn+1, c; f/x)(10)

=

[Rn+1(x)

Rn+1(x)

]2f(c) + (x− c)

n+1∑k=1

φk(x)

xk − xBk(f ;x),

wherex1, x2, ..., xn+1 are the roots of Rn+1(x),c ∈ [−1, 1], c = xk, k = 1, n+ 1

φk(x) =

[Rn+1(x)

(x− xk)R′n+1(xk)

]2Bk(f ;x) = f(xk) + (x− xk)

[f ′(xk)−

R′′n+1(xk)

R′n+1(xk)

f(xk)−f(xk)

xk − c

].

Further, we investigate Bk(f ;x).From (9) we obtain

(11) f(xk) = (n+ 1)R2n(xk).

We have

f ′(x) = 2(n+ 1)Rn(x)R′n(x)−Rn(x)Rn+1(x)− xR′

n(x)Rn+1(x)− xRn(x)R′(n+1)(x)

− nR′n−1(x)Rn+1(x)− nRn−1(x)R

′(n+1)(x),

f ′(xk) = 2(n+ 1)Rn(xk)R′n(xk)− xkRn(xk)R

′n+1(xk)− nRn−1(xk)R

′n+1(xk).

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Characterization theorems of Jacobi and Laguerre polynomials 121

Using (2), (3) we find

Rn−1(xk) =2n+ 2α+ 1

nxkRn(xk),

R′n(xk) = (n+ 2α+ 1)

xk1− x2k

Rn(xk),

f ′(xk) = 2α(n+ 1)xk

1− x2kR2

n(xk).

For c = 1, observe that

Bk(f ;x) = (n+ 1)1 + x

1 + xkR2

n(xk), f(1) = 0,

(12) f(x) =(1− x2)R2

n+1(x)

n+ 1

n+1∑k=1

1− x2k(x− xk)2

.

From (2), (3), (9) and (12) we obtain

nRn−1(x)

Rn+1(x)= (2n+ 2α+ 1)x

Rn(x)

Rn+1(x)− (n+ 2α+ 1),

(n+ 1)

[Rn(x)

Rn+1(x)

]2− x

Rn(x)

Rn+1(x)− n

Rn−1(x)

Rn+1(x)=

1− x2

n+ 1

n∑k=1

1− x2k(x− xk)2

,

(n+ 1)

[Rn(x)

Rn+1(x)

]2− (2n+ 2α+ 2)x

Rn(x)

Rn+1(x)+ (n+ 2α+ 1) =

1− x2

n+ 1

n∑k=1

1− x2k(x− xk)2

,

Rn(x)

Rn+1(x)= x+

1− x2

n+ 1·R′

n+1(x)

Rn+1(x),

R′n+1(x)

Rn+1(x)=

n+1∑k=1

1

x− xk,

(n+ 2α+ 1)(1− x2)− 2αx1− x2

n+ 1·n+1∑k=1

1

x− xk+

(1− x2)2

n+ 1

n+1∑k=1

1

(x− xk)2

+ 2(1− x2)2

n+ 1

∑1≤i<k≤n+1

1

(x− xi)(x− xk)2=

(1− x2)2

n+ 1

n∑k=1

1

(x− xk)2

+2x(1− x2)

n+ 1

n∑k=1

1

x− xk− (1− x2),

n+ 2α+ 2− 2(α+ 1)x

n+ 1

n+1∑k=1

1

x− xk+

2(1− x2)

n+ 1

∑1≤i<k≤n+1

1

(x− xi) · (x− xk)= 0.

In conclusion, we proof the following theorems:

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122 I. Tincu

Theorem 1 If x1, x2, ..., xn+1 ⊂ R, xi = xj for i = j, i, j ∈ 1, 2, ..., n + 1verifies

(13)1− x2

n+ 1

∑1≤i<k≤n+1

1

(x− xi)(x− xk)− α+ 1

n+ 1x

n+1∑k=1

1

x− xk+n+ 2α+ 2

2= 0,

x ∈ (−1, 1), x = xk, k = 1, n+ 1 then xk ∈ (−1, 1), k = 1, n+ 1 (or R(α)n+1(xk) =

0, k = 1, n+ 1).

Theorem 2 If x1, x2, ..., xn ⊂ R, xi = xj for i = j, i, j ∈ 1, 2, ..., n,xj = 0, j ∈ 1, 2, ..., n verifies

(14)xj − α− 1

2xj=

n∑k=1k =j

1

xj − xk,

for α > −1 then xj > 0, (∀)j = 1, n.

Proof. Let P (x) =n∏

k=1

(x− xk). We obtain

P ′(x)

P (x)− 1

x− xj=

n∑k=1k =j

1

x− xk,

limx→xj

(x− xj)P′(x)− P (x)

(x− xj)P (x)=

n∑k=1k =j

1

xj − xk,

(15)∑k=1k =l

1

xj − xk=

P ′′(xj)

2P ′(xj).

From (14) and (15) we have

P ′′(xj)

2P ′(xj)=xj − α− 1

2xj,

(16) xjP′′(xj) = (xj − α− 1)P ′(xj), j ∈ 1, 2, ..., n.

Let h(x) = xP ′′+(α+1−x)P ′(x). From (16), we observe h(xj) = 0, j = 1, 2, ..., n.In conclusion exists cn ∈ R∗ such that h(x) = cnP (x), then

(17) xP ′′(x) + (α+ 1− x)P ′(x)− cnP (x) = 0.

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Characterization theorems of Jacobi and Laguerre polynomials 123

Because L(α)0 (x), L

(α)1 (x), ..., L

(α)n (x) is base in Πn, exists bk ∈ R, k ∈ 0, 1, ..., n

such that

P (x) =

n∑k=0

bkL(α)k (x).

From (4) and (17) we obtain

bk = 0, k ∈ 0, 1, ..., n− 1, cn = −n.

In conclusion, the polynomial P verifies following identity

xP ′′(x) + (α+ 1− x)P ′(x) + nP (x) = 0,

hence it exists λn ∈ R such that P (x) = λnL(α)n (x).

Therefore xj > 0, j − 1, n.

References

[1] G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Providence, R.I.1985

[2] I. Tincu, A property of Laguerre polynomials, General Mathematics, Vol.14,No.4, 2006

[3] I. Tincu, Characterization theorems of Jacobi polynomials, General Mathe-matics, Vol.20, No.5, 2012

Ioan TincuLucian Blaga University of SibiuDepartment of Mathematics and InformaticsStr. Dr. I. Ratiu, No.5-7RO-550012 Sibiu, Romaniae-mail: [email protected]

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General Mathematics Vol. 22, No. 1 (2014), 125–131

Interpolation operators on a triangle with one curvedside 1

Alina Babos

Dedicated to the late Academician Professor Dr. Dimitrie D. Stancu

Abstract

We construct Hermite and Birkhoff-type operators, which interpolate a givenfunction and some of its derivatives on some interior lines of a triangle with onecurved side. We consider the case when the interior line is a median. We alsoconsider some of their product and Boolean sum operators. We study the in-terpolation properties and the degree of exactness of the constructed operators.

2010 Mathematics Subject Classification: 65D30, 65D32, 26A15.Key words and phrases: Triangle, curved edges, interpolation operators.

1 Introduction

There have been constructed interpolation operators of Lagrange, Hermite andBirkhoff type on a triangle with all straight sides, starting with the paper [5] ofR.E. Barnhil, G. Birkhoff and W.J. Gordon, and in many others papers (see, e.g.,[4], [6], [7], [10], [11]). Then, were considered interpolation operators on triangleswith curved sides (one, two or all curved sides), many of them in connection withtheir applications in computer aided geometric design and in finite element analysis(see, e.g, [1], [2], [8],[9], [12]-[15]).

In [12] the authors consider a standard triangle,Th, having the verticesV1 = (h, 0), V2 = (0, h) and V3 = (0, 0), two straight sides Γ1,Γ2, along the co-ordinate axes, and the third side Γ3 (opposite to the vertex V3), which is definedby the one-to-one functions f and g, where g is the inverse of the function f, i.e.y = f(x) and x = g(y), with f(0) = g(0) = h and F a real-valued function definedon Th (see Figure 1).

1Received 5 June, 2014Accepted for publication (in revised form) 3 August, 2014

125

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126 A. Babos

Figure 1.

They construct certain Lagrange, Hermite and Birkhoff type operators, whichinterpolate a given function and some of its derivatives on the border of this trianglewith one curved side, as well as some of their product and Boolean sum operators.

In [1] we introduce an Lagrange operator which interpolate the function F oncathetus, on the curved side, but also on a interior line of the triangle Th. We con-sidered the case when the interior line is a median. Thus we have introduced theoperator Lx

2 which interpolate the function F in relation to x in points (0, y),(h−y2 , y

)and (g(y), y):

(Lx2F )(x, y) =

(2x− h+ y)[x− g(y)]

(h−y)g(y)F (0, y)+

4x[x− g(y)]

(h−y)[h− y −2g(y)]F

(h− y

2, y

)

+x(2x− h− y)

g(y)[2g(y)− h+ y]F (g(y), y)).

So, the operator Lx2 interpolate the function F on cathetus V2V3, on curved side and

on median V2M (see Figure 2).

Figure 2.

In this paper we construct Hermite and Birkhoff-type operators, which interpo-late a given function and some of its derivatives on the median. We also considersome of their product and Boolean sum operators. We study the interpolation

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Interpolation operators on a triangle with one curved side 127

properties and the degree of exactness of the constructed operators. Recall thatdex(P ) = r (where P is an interpolation operators) if Pf = f , for f ∈ Pr, and thereexists g ∈ Pr+1 such that Pg = g, where Pm denote the space of the polynomials intwo variables of global degree at most m.

2 Hermite and Birkhoff-type operators

Let H1 be the Hermite operator who interpolate the function F with respect to x

in the points (0, y),(h−y2 , y

), (g(y), y) :

(H1F )(x, y) =(2x− h+ y)2[x− g(y)]2

(h− y)2g2(y)F (0, y)

+16x[x− g(y)]2[2(h− y− g(y))(h− y− x)− x(h− y)]

(h− y)2[h− y− 2g(y)]3F(h− y

2, y)

+4x[x− g(y)]2(2x− h+ y)

(h− y)[h− y − 2g(y)]2F (1,0)

(h− y

2, y)

(1)

+x(2x− h+ y)2[2g(y)(4g(y)− h+ y)+ x(h− y − 6g(y))]

g2(y)[2g(y)− h+ y]3F (g(y), y)

+x[x− g(y)](2x− h+ y)2

g(y)[2g(y)− h+ y]2F (1,0)(g(y), y).

Theorem 1 If F : Th → R, then we get

1. the interpolation properties:

H1F = F, onΓ1 ∪ Γ3 ∪ V2M,

H1F(1,0) = F (1,0), onΓ3 ∪ V2M,

(see Figure 3).

2. the degree of exactness: dex(H1) = 2.

Figure 3.

Page 128: 2014 Volume 22 No. 1

128 A. Babos

Proof. 1.(H1F )(0, y) = F (0, y),

(H1F )(h− y

2, y)= F

(h− y

2, y), (H1F )(g(y), y) = F (g(y), y),

(H1F )(1,0)

(h− y

2, y)= F (1,0)

(h− y

2, y), (H1F )

(1,0)(g(y), y) = F (1,0)(g(y), y).

2. We obtain H1eij = eij for i+ j ≤ 2 and H1e30 = e30, where eij(x, y) = xiyj . So,it follows that dex(H1) = 2.

Let H2 be the interpolation operator defined in [12]:

(H2F ) =[y − f(x)]2

f2(x)F (x, 0) +

y[2f(x)− y]

f2(x)F (x, f(x))

+y[y − f(x)]

f(x)F (0,1)(x, f(x)).

The product of the operators H1 and H2 is given by

(P21F )(x, y) =[y − f(x)]2

f2(x)

[(2x− h)2(x− h)2

h4F (0, 0) +

16x2(x− h)2

h4F(h2, 0)

+4x(x− h)2(2x− h)

h3F (1,0)

(h2, 0)

+x(2x− h)2(6h− 5x)

h4F (h, 0) +

x(x− h)(2x− h)2

h3F (1,0)(h, 0)

]+

y[2f(x)− y]

f2(x)F (x, f(x)) +

y[y − f(x)]

f(x)F (0,1)(x, f(x))

with the interpolation properties:

P21F = F, onΓ2 ∪ Γ3,

(P21F )(1,0) = F (1,0), onΓ2, (P21F )

(1,0) = F (0,1), onΓ3,

(see Figure 4) and the degree of exactness: dex(P21) = 2.

Figure 4.

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Interpolation operators on a triangle with one curved side 129

The boolean sum of the operators H1 and H2 is given by

(S21F )(x, y) = H2 ⊕H1 = H2 +H1 −H2 ·H1

=[y − f(x)]2

f2(x)F (x, 0) +

(2x− h+ y)2[x− g(y)]2

(h− y)2g2(y)F (0, y)

+16x[x− g(y)]2[2(h− y − g(y))(h− y − x)− x(h− y)]

(h− y)2[h− y − 2g(y)]3F(h− y

2, y)

+4x[x− g(y)]2(2x− h+ y)

(h− y)[h− y − 2g(y)]2F (1,0)

(h− y

2, y)

+x(2x− h+ y)2[2g(y)(4g(y)− h+ y) + x(h− y − 6g(y))]

g2(y)[2g(y)− h+ y]3F (g(y), y)

+x[x− g(y)](2x− h+ y)2

g(y)[2g(y)− h+ y]2F (1,0)(g(y), y)

− [y − f(x)]2

f2(x)

[(2x− h)2(x− h)2

h4F (0, 0) +

16x2(x− h)2

h4F(h2, 0)

+4x(x− h)2(2x− h)

h3F (1,0)

(h2, 0)

+x(2x− h)2(6h− 5x)

h4F (h, 0) +

x(x− h)(2x− h)2

h3F (1,0)(h, 0)

]with the interpolation properties:

S21F = F, on ∂Th ∪ V2M,

(S21F )(1,0) = F (1,0), onΓ3 ∪ V2M,

and the degree of exactness: dex(SH21) = 2.

We suppose that the function F : Th → R has the partial derivatives F (1,0) onΓ3 and V2M .

We consider the Birkhoff-type operator B1 defined by

(B1F )(x, y) = F (0, y) +x[2g(y)− x]

2g(y)− h+ yF (1,0)

(h− y

2, y)

(2)

+x(x− h+ y)

2g(y)− h+ yF (1,0)(g(y), y).

Theorem 2 If F : Th → R, then we get

1. the interpolation properties:

B1F = F, onΓ1,

(B1F )(1,0) = F (1,0)onΓ3 ∪ V2M,

(see Figure 5).

Page 130: 2014 Volume 22 No. 1

130 A. Babos

2. the degree of exactness: dex(B1) = 2.

Figure 5.

Proof. 1.

(B1f)(0, y) = F (0, y)

(B1F )(1,0)

(h− y

2, y)= F (1,0)

(h− y

2, y), (B1F )

(1,0)(g(y), y) = F (1,0)(g(y), y).

2. B1eij = eij for i+j ≤ 2 and B1e30 = e30, where eij(x, y) = xiyj . So, it followsthat dex(B1) = 2.

References

[1] A. Babos, Some interpolation operators on triangle, The 16th InternationalConference The Knowledge - Based Organization, Applied Technical Sciencesand Advanced Military Technologies, Conference Proceedings 3, Sibiu, 28-34,2010.

[2] A. Babos, Some interpolation schemes on a triangle with one curved side,General Mathematics (accepted).

[3] A. Babos, Interpolation operators on a triangle with two and three curved edges,Creat. Math. Inform. 22, No. 2, 135-142, 2013.

[4] R.E. Barnhil, J.A. Gregory, Polynomial interpolation to boundary data ontriangles, Math. Comput. 29(131), 726-735, 1975.

[5] R.E. Barnhil, G. Birkoff, W.J. Gordon, Smooth interpolation in triangle, J.Approx. Theory. 8, 114-128, 1973.

[6] R.E. Barnhil, L. Mansfield, Error bounds for smooth interpolation, J. Approx.Theory. 11,306-318, 1974.

[7] D. Barbosu, I. Zelina, About some interpolation formulas over triangles, Rev.Anal. Numer. Theor. Approx, 2, 117-123, 1990.

Page 131: 2014 Volume 22 No. 1

Interpolation operators on a triangle with one curved side 131

[8] D. Barbosu, Aproximarea functiilor de mai multe variabile prin sume booleenede operatori liniari de tip interpolator, Ed.Risoprint, Cluj-Napoca, 2002.

[9] C. Bernardi, Optimal finite-element interpolation on curved domains, SIAM J.Numer. Anal., 26, 1212-1240, 1989.

[10] G. Birkhoff, Interpolation to boundary data in triangles, J. Math. Anal. Appl.,42, 474-484, 1973.

[11] T. Catinas, Gh. Coman, Some interpolation operators on a simplex domain,Studia Univ. Babes Bolyai, LII, 3, 25-34, 2007.

[12] Gh. Coman, T. Catinas, Interpolation operators on a triangle with one curvedside, BIT Numer Math 47, 2010.

[13] Gh. Coman, T. Catinas, Interpolation operators on a tetrahedron with threecurved sides, Calcolo, 2010.

[14] W.J. Gordon, Ch. Hall, Transfinite element methods: blending-function inter-polation over arbitrary curved element domains Numer. Math. 21, 109-129,1973.

[15] J.A. Marshall, R. McLeod, Curved elements in the finite element method,Conference on Numer. Sol. Diff. Eq., Lectures Notes in Math., 363, SpringerVerlag, 89-104, 1974.

Alina Babos”Nicolae Balcescu” Land Forces AcademySibiu, Romaniae-mail: alina babos [email protected]

Page 132: 2014 Volume 22 No. 1
Page 133: 2014 Volume 22 No. 1

General Mathematics Vol. 22, No. 1 (2014), 133–138

Some inequalities for the Landau constants 1

Emil C. Popa

Dedicated to the late Academician Professor Dr. Dimitrie D. Stancu

Abstract

In this paper we obtain some new inequalities for the Landau constants.

2010 Mathematics Subject Classification: 26D15, 26D20, 33C20, 33B15.Key words and phrases: Wallis inequality, Zhang inegualities, Landau

constants, Riemann Zeta function.

1 Introduction and results

Let An =(2n− 1)!!

(2n)!!, n = 1, 2, . . . directly related to the Wallis formula

(1) limn→∞

1

nA2n

= π

and the Wallis inequalities

2

2n+ 1< πA2

n <1

n, n = 1, 2, . . . .

G.M. Zhang [6] has proved the double inequality

(2)32n+ 8

32n2 + 16n+ 3< πA2

n <8n+ 3

8n2 + 5n+ 1, n = 1, 2, . . .

Theorem 1 For n ∈ N, n ≥ 1 we have

(3)32n2 + 16n+ 3

32n3 + 24n2 + 8n+ 1< πA2

n.

1Received 9 July, 2014Accepted for publication (in revised form) 1 September, 2014

133

Page 134: 2014 Volume 22 No. 1

134 E.C. Popa

Theorem 2 For n ∈ N, n ≥ 3 we have

(4) πA2n <

8n2 + 35n+ 8

8n3 + 37n2 + 17n+ 3.

Remark 1 We observe that

32n+ 8

32n2 + 16n+ 3<

32n2 + 16n+ 3

32n3 + 24n2 + 8n+ 1< πA2

n <

8n2 + 35n+ 8

8n3 + 37n2 + 17n+ 3<

8n+ 3

8n2 + 5n+ 1

Hence (3) is an improvement for the lower bound in Zhang’s inequality (2) and theinequality (4) is an improvement for the upper bound in Zhang’s inequality (2).

The sums

Gn =n∑

k=0

1

16k

(2k

k

)2

, n = 0, 1, 2, . . .

are known as the Landau constants. It was proved by Landau in 1913, that, if afunction f(z) such that |f(z)| < 1 for |z| < 1, is analytic throughout the interior ofthe unit circle and its Taylor series is

f(z) =

∞∑i=0

aizi, then

∣∣∣∣∣n∑

i=0

ai

∣∣∣∣∣ ≤ Gn.

Moreover, if Tn(f) is the Taylor polynom associated to f(z), then its norm is givenby ||Tn|| = Gn.

D. Zhao establishes in [7] several inequalities for Gn.

(5)

1

πln(n+ 1) + c0 −

1

4π(n+ 1)+

5

192π(n+ 1)2< Gn

Gn <1

πln(n+ 1) + c0 −

1

4π(n+ 1)+

5

192π(n+ 1)2+

3

128π(n+ 1)3,

where n ≥ 1, c0 =1

π(γ + 4 ln 2) = 1, 06627 . . . , and γ = 0, 57721 . . . denotes Euler’s

constant.Using the Riemann Zeta function, we obtained in [4] and [5] some improvements

of inequalities (5) in the following evaluations

(6)

1

πln(n+1)+c0−

1

4π(n+1)+

5

192π(n+1)2+

17

256π

(ζ(4)−

n+1∑k=1

1

k4

)<Gn

Gn <1

πln(n+ 1) + c0 −

1

4π(n+ 1)+

5

192π(n+ 1)2

+9

128π

(ζ(4)−

n∑k=1

1

k4

)− 2263

61440π(n+ 1)4,

Page 135: 2014 Volume 22 No. 1

Some inequalities for the Landau constants 135

where ζ(s) =

∞∑k=1

1

ksis the Riemann - Zeta function.

In this paper we will establish some sharp inequalities for Gn wich improve theinequalities (6).

Theorem 3 We have for n ≥ 3

(7)

1

πln(n+ 2) + c0 −

1

4π(n+ 2)+

5

192π(n+ 2)2

+17

256π

(ζ(4)−

n+2∑k=1

1

k4

)− 1

π· 8n2 + 51n+ 51

8n3 + 61n2 + 115n+ 65< Gn,

and for n ≥ 1(8)

Gn <1

πln(n+ 2) + c0 −

1

4π(n+ 2)+

5

192π(n+ 2)2

+9

128π

(ζ(4)−

n+1∑k=1

1

k4

)− 2263

61440π(n+ 2)4− 1

π· 32n2 + 80n+ 51

32n3 + 120n2 + 152n+ 65.

2 Proofs of the theorems

We are now able to prove our theorems.Proof of Theorem 1. We denote

αn = A2n

32n3 + 24n2 + 8n+ 1

32n2 + 16n+ 3.

From Wallis formula (1) it follows that

(9) limn→∞

αn =1

π.

We have now

αn+1 − αn = A2n

((2n+ 1)2(32n3 + 120n2 + 152n+ 65)

(2n+ 2)2(32n2 + 80n+ 51)− 32n3 + 24n2 + 8n+ 1

32n2 + 16n+ 3

)= −A2

n

56n2 + 84n+ 9

(2n+ 2)2(32n2 + 80n+ 51)(32n2 + 16n+ 3)< 0.

It is obvious that αn+1 < αn for any n ∈ N, n ≥ 1. Next, by using (9) we deduce

that αn >1

πand hence

πA2n >

32n2 + 16n+ 3

32n3 + 24n2 + 8n+ 1, n ≥ 1.

Page 136: 2014 Volume 22 No. 1

136 E.C. Popa

Proof of Theorem 2. We denote

βn = A2n

8n3 + 37n2 + 17n+ 3

8n2 + 35n+ 8.

We have

βn−1 − βn = A2n

((2n+ 1)2(8n3 + 61n2 + 115n+ 65)

(2n+ 2)2(8n2 + 51n+ 51)− 8n3 + 37n2 + 17n+ 3

8n2 + 35n+ 8

)= A2

n

3n3 + 9n2 − 29n− 92

(2n+ 2)2(8n2 + 51n+ 51)(8n2 + 35n+ 8)> 0.

and hence βn+1 > βn for any n ≥ 4.

By using (9) we deduce that βn <1

π, or

πA2n <

8n2 + 35n+ 8

8n3 + 37n2 + 17n+ 3, for n ≥ 3.

Proof of Theorem 3. It is easy to verify that

(10) Gn = Gn+1 −A2n+1

From (3) and (4) we obtain

(11)1

π

32n2 + 80n+ 51

32n3 + 120n2 + 152n+ 65< A2

n+1 <1

π

8n2 + 51n+ 51

8n3 + 61n2 + 115n+ 65.

Using (6) and (11), we obtain from (10) the inequalities (7) and (8).

Remark 2 For n ∈ N, n ≥ 1 we have

(12)

8n2 + 51n+ 51

8n3 + 61n2 + 115n+ 65< ln

(1 +

1

n+ 1

)+

1

4

(1

n+ 1− 1

n+ 2

)

− 5

192

(1

(n+ 1)2− 1

(n+ 2)2

)− 17

256(n+ 2)2.

Proof. From [4, p. 115-116] we have

8n+ 3

8n2 + 5n+ 1< ln

(1 +

1

n

)+

1

4

(1

n− 1

n+ 1

)− 5

192

(1

n2− 1

(n+ 1)2

)− 17

256(n+ 1)2.

But8n2 + 51n+ 51

8n3 + 61n2 + 115n+ 65<

8n+ 11

8n2 + 21n+ 14

< ln

(1 +

1

n+ 1

)+1

4

(1

n+ 1− 1

n+ 2

)− 5

192

(1

(n+ 1)2− 1

(n+ 2)2

)− 17

256(n+ 2)2,

and the inequality (12) follows.Notice that, according to Remark 2 the first inequality (7) is an improvment of

the first inequality from (6).

Page 137: 2014 Volume 22 No. 1

Some inequalities for the Landau constants 137

Remark 3 For n ≥ 1 we have

32n2 + 80n+ 51

32n3 + 120n2 + 152n+ 65> ln

(1 +

1

n+ 1

)+

1

4

(1

n+ 1− 1

n+ 2

)

− 5

192

(1

(n+ 1)2− 1

(n+ 2)2

)− 9

128(n+ 1)4+

2263

61440

(1

(n+ 1)4− 1

(n+ 2)4

).

Proof. From [5, p. 1459] we obtain

(13)

32n+ 8

32n2 + 16n+ 8> ln

(1 +

1

n

)+

1

4

(1

n− 1

n+ 1

)

− 5

192

(1

n2− 1

(n+ 1)2

)− 9

128n4+

2263

61440

(1

n4− 1

(n+ 1)4

).

But,32n2 + 80n+ 51

32n3 + 120n2 + 152n+ 65>

32n+ 40

32n2 + 80n+ 56

> ln

(1 +

1

n+ 1

)+

1

4

(1

n+ 1− 1

n+ 2

)− 5

192

(1

(n+ 1)2− 1

(n+ 2)2

)− 9

128(n+ 1)4+

2263

61440

(1

(n+ 1)4− 1

(n+ 2)4

),

and the inequality (13) follows.

Notice that, according to Remark 3 the second inequality (8) is an improvementof the second inequality (6).

References

[1] L. Brutman, A sharp estimate of the Landau constants, J. Aprox. Theory, 34(1982), 217-220.

[2] C.P. Chen, Approximation formulas for Landau’s constants, J. Math. Anal.Appl., 387 (2012), 916-919.

[3] C.P. Chen, F. Qi, H. Alzer, The best bounds in Wallis inequality, Proc. Amer.Math. Soc., 133, no. 2, (2005), 113-117.

[4] E.C. Popa, Note of the constants of Landau, Gen. Math., vol. 18 (2010), no. 1,113-117.

[5] E.C. Popa, N.A. Secelean On some inequality for the Landau constants,Taiwanese J. Math., vol. 15, no. 4, (2011), 1457-1462.

Page 138: 2014 Volume 22 No. 1

138 E.C. Popa

[6] G.M. Zhang, The upper bounds and lower bounds on Wallis inequality, Math.Pract. Theory, 37(5) (2007), 111-116 (Chinese).

[7] D. Zhao, Some sharp estimates of the constants of Landau and Lebesque, J.Math. Anal. Appl., 349 (2009), 68-73.

Emil C. PopaLucian Blaga University of SibiuDepartment of Mathematics and InformaticsStr. Dr. I. Ratiu, No.5-7RO-550012 Sibiu, Romaniae-mail: [email protected]

Page 139: 2014 Volume 22 No. 1

General Mathematics Vol. 22, No. 1 (2014), 139–149

Error bounds for a class of quadrature formulas 1

Florin Sofonea, Ana Maria Acu, Arif Rafiq, Dan Barbosu

Dedicated to the late Academician Professor Dr. Dimitrie D. Stancu

Abstract

A class of optimal quadrature formulas in sense of minimal error boundsare obtained. The estimations of remainder term will be given in terms ofvariety of norms, from an inequality point of view. Some improvements andgeneralizations of some results from literature will be considered.

2010 Mathematics Subject Classification: 65D30, 65D32, 26A15.Key words and phrases: optimal quadrature formula, corrected quadrature

formula, remainder term.

1 Introduction

In the last years the problem of numerical integration attracted attention of manyauthors. The deduction of the optimal quadrature formulas, in terms of varietyof norms, from an inequality point of view was considered by Ujevic ([7]-[10]) whoobtained optimal two-point and three-point quadrature formulas. In [12], F. Zafarand N.A. Mir found out an optimal quadrature formula of the form∫ 1

−1f(t)dt− [hf(−1) + (1− h)f(x) + (1− h)f(y) + hf(1)](1)

=

∫ 1

−1K(x, y, t)f ′′(t)dt, where K(x, y, t) is defined by

(2) K(x, y, t) =

1

2(t− α)2 + α1, t ∈ [−1, x],

1

2(t− β)2 + β1, t ∈ (x, y),

1

2(t− γ)2 + γ1, t ∈ [y, 1],

1Received 15 June, 2014Accepted for publication (in revised form) 27 August, 2014

139

Page 140: 2014 Volume 22 No. 1

140 F. Sofonea, A. M. Acu, A. Rafiq, D. Barbosu

and x, y ∈ [−1 + 2h, 1− 2h] with x < y, h ∈[0, 12].

The parameters α, α1, β, β1, γ, γ1 ∈ R involved in K(x, y, t) are required to bedetermined in a way such that the representation (1) is obtained.

The nodes x and y are obtained putting conditions that the remainder termwhich is evaluated in the following sense∣∣∣∣∫ 1

−1K(x, y, t)f ′′(t)dt

∣∣∣∣ ≤ maxt∈[a,b]

|f ′′(t)|∫ 1

−1|K(x, y, t)|dt

to be minimal, namely

∫ 1

−1|K(x, y, t)|dt to attains the minimum value.

F. Zafar and N.A. Mir found the quadrature formula of type (1) such that theestimation of its error to be best possible in uniform norm. The main result obtainedby F. Zafar and N.A. Mir in the above described procedure is formulated bellow

Theorem 1 [12] Let I ⊂ R be an open interval such that [−1, 1] ⊂ I and letf : I → R be a twice differentiable function such that f ′′ is bounded and integrable.Then, ∫ 1

−1f(t)dt =

[hf(−1) + (1− h)f(−4 + 4h+ 2

√3− 6h+ 4h2)(3)

+ (1− h)f(4− 4h− 2√

3− 6h+ 4h2) + hf(1)]+R[f ],

where |R[f ]| ≤ 2∆(h) ∥f ′′∥∞ , h ∈[0,

1

2

], and∆(h) is defined as

∆(h) =52

3h3 − 44h2 +

83

2h− 83

6+ 8(1− h)2

√4h2 − 6h+ 3

+2

3

[8h2 − 14h+ 7− 4(1− h)

√4h2 − 6h+ 3

] 32.

In [1] using a similar way by F. Zafar and A. Mir is obtained the following optimalquadrature formula, where the estimation of the error is best possible in L2-norm.

Theorem 2 [1] Let I ⊂ R be an open interval such that [−1, 1] ⊂ I and let f : I →R be a twice differentiable function such that f ′′ is integrable. Then,∫ 1

−1f(t)dt = hf(−1) + (1− h)f(−3h+ 3−

√9h2 − 12h+ 6)(4)

+ (1− h)f(3h− 3 +√

9h2 − 12h+ 6) + hf(1) +R[f ],

where |R[f ]| ≤ Φ(h) ∥f ′′∥2 , h ∈(−∞,

1

2

], and Φ(h) is defined as

Φ(h) =

−36h5 + 144h4 − 240h3 +

632

3h2 − 98h+

98

5

+[−12h4 + 40h3 − 52h2 + 32h− 8

]√9h2 − 12h+ 6

12.

Page 141: 2014 Volume 22 No. 1

Error bounds for a class of quadrature formulas 141

2 A class of optimal quadrature formulas

The quadrature formulas (3) and (4) can be obtained using Peano’s theorem. In thissection we will present this method. Also, an optimal quadrature formula, wherethe estimation of the error is best possible in L1-norm will be obtained.

We consider the quadrature formula (1) has degree of exactness equal 1. Sincethe quadrature formula has degree of exactness 1, the remainder term verifies theconditions R[ei] = 0, ei(x) = xi, i = 0, 1, and we obtain y = −x. We can choosex ≥ 0.

Let I ⊂ R be an open interval such that [−1, 1] ⊂ I and let f : I → R be atwice differentiable function such that f ′′ is integrable. Using Peano’s theorem theremainder term has the following integral representation

(5) R[f ] =

∫ 1

−1K(t)f ′′(t)dt, where

K(t) = R[(x− t)+] =

t2

2+ t(1− h)− h+

1

2, t ∈ [−1,−x],

t2

2− (1− h)x− h+

1

2, t ∈ (−x, x),

t2

2− t(1− h)− h+

1

2, t ∈ [x, 1].

For the remainder term we have the evaluation

|R[f ]| ≤[∫ 1

−1|f ′′(t)|pdt

] 1p[∫ 1

−1|K(t)|qdt

] 1q

,1

p+

1

q= 1,

with the remark that in the cases p = 1 and p = ∞ this evaluation is

|R[f ]| ≤∫ 1

−1|f ′′(t)|dt · sup

t∈[−1,1]|K(t)|,(6)

|R[f ]| ≤ supt∈[−1,1]

|f ′′(t)| ·∫ 1

−1|K(t)|dt.(7)

The function

∫ 1

−1|K(t)|dt attains the minimum value if K|[−x,x] is the Chebyshev

orthogonal polynomial of the second kind of degree 2, on the interval [−1, 1] with

the coefficient of t2 equal to1

2, namely

(8)t2

2− (1− h)x− h+

1

2=

1

2x2U2

(t

x

),

where U2(t) = t2 − 1

4is the Chebyshev polynomial of the second kind of degree 2,

on the interval [−1, 1]. Ecuation (8) has the following solutions

x1 = 4(1− h)− 2√

4h2 − 6h+ 3,

x2 = 4(1− h) + 2√

4h2 − 6h+ 3.

Page 142: 2014 Volume 22 No. 1

142 F. Sofonea, A. M. Acu, A. Rafiq, D. Barbosu

We find that x1 is the global minima of F (x) =

∫ 1

−1|K(t)|dt and the result of F.Zafar

and N.A. Mir is obtained.

The function

∫ 1

−1(K(t))2 dt attains the minimum value ifK|[−x,x] is the Legendre

orthogonal polynomial of degree 2, on the interval [−x, x], with the coefficient of t2

equal to1

2, namely

(9)t2

2− (1− h)x− h+

1

2=

1

2x2X2

(t

x

),

where X2(t) = t2 − 1

3is the Legendre orthogonal polynomial of degree 2, on the

interval [−1, 1]. Equation (9) has the following solutions

x1 = 3(1− h)−√

9h2 − 12h+ 6,

x2 = 3(1− h) +√

9h2 − 12h+ 6.

We find that x1 is the global minima of G(x) =

∫ 1

−1(K(t))2 dt and Theorem 2 is

obtained.In the following Theorem we give a minimal estimation of the error bound in L1-

norm for the quadrature formula (1). In order to obtain this result we will considerthe estimation (6).

Theorem 3 Let I ⊂ R be an open interval such that [−1, 1] ⊂ I and let f : I → Rbe a twice differentiable function such that f ′′ is integrable. Then,∫ 1

−1f(t)dt = hf(−1) + (1− h)f

(2− 2h−

√4h2 − 4h+ 2

)(10)

+ (1− h)f(−2 + 2h+

√4h2 − 4h+ 2

)+ hf(1) +R[f ],

where |R[f ]| ≤ Ψ(h)∥f ′′∥1, h ∈[0,

1

2

], and Ψ(h) is defined as

Ψ(h) =

3

2− 3h+ 2h2 + (h− 1)

√4h2 − 4h+ 2, h ∈

[0,−1

7+

2

7

√2

]h2

2, h ∈

(−1

7+

2

7

√2,

1

2

].

Proof. The function supt∈[−1,1]

|K(t)| attains the minimum value if K|[−x,x] is the

Chebyshev orthogonal polynomial of the first kind of degree 2, on the interval [−x, x],with the coefficient of t2 equal to

1

2, namely

(11)t2

2− (1− h)x− h+

1

2=

1

2x2T2

(t

x

),

Page 143: 2014 Volume 22 No. 1

Error bounds for a class of quadrature formulas 143

where T2(t) = t2 − 1

2is the Chebyshev orthogonal polynomial of the first kind of

degree 2, on the interval [−1, 1]. Equation (11) has the following solutions

x1 = 2− 2h−√

4h2 − 4h+ 2,

x2 = 2− 2h+√

4h2 − 4h+ 2.

Denote H(x) = supt∈[−1,1]

|K(t)|. We find that H(x1) < H(x2), therefore

x1 = 2− 2h−√

4h2 − 4h+ 2

is the global minima of H(x).

Remark 1 For h = −1

7+

2

7

√2, Ψ(h) attains its minimum value.

Corollary 1 Let the assumption of Theorem 3 holds. Then, one has followingoptimal quadrature rule∫ 1

−1f(t)dt =

1

7

(2√2− 1)f(−1) + (8− 2

√2)f

(4−

√2

7

)

+ (8− 2√2)f

(√2− 4

7

)+ (2

√2− 1)f(1) +R[f ]

,

where |R[f ]| ≤ (2√2− 1)2

98∥f ′′∥1.

Theorem 4 For the remainder term of the quadrature formula (10) the followingestimations can be established

(i) |R[f ]| ≤ η(h)∥f ′′∥∞, where

η(h) =32

3h3 − 20h2 +

33

2h− 11

2+ 4(h− 1)2

√4h2 − 4h+ 2

+ 4

(4

3h2 − 2h+ 1 +

2

3(h− 1)

√4h2 − 4h+ 2

)·√

3− 6h+ 4h2 + 2(h− 1)√

4h2 − 4h+ 2;

(ii) |R[f ]| ≤ ν(h)∥f ′′∥2, where

ν(h) =

−64

3h5 + 80h4 − 376

3h3 +

314

3h2 − 283

6h+

283

30

+√

4h2 − 4h+ 2

(80

3h− 44h2 +

104

3h3 − 32

3h4 − 20

3

)1/2

.

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144 F. Sofonea, A. M. Acu, A. Rafiq, D. Barbosu

Proof. The inequality (i) follows considering the following estimation of remainderterm

|R[f ]| ≤ ∥f ′′∥∞∫ 1

−1|K(t)|dt.

Denote

K1(t) :=t2

2+ t(1− h)− h+

1

2;

K2(t) :=t2

2− (1− h)x− h+

1

2;

K3(t) :=t2

2− t(1− h)− h+

1

2.

We have∫ 1

−1|K(t)|dt =

∫ 2h−1

0−K1(t)dt+

∫ −(2−2h−√4h2−4h+2)

2h−1K1(t)dt

+

∫ −√

3−6h+4h2+2(h−1)√4h2−4h+2

−(2−2h−√

4h2−4h+2)K2(t)dt+

∫ √3−6h+4h2+2(h−1)

√4h2−4h+2

−√

3−6h+4h2+2(h−1)√4h2−4h+2

−K2(t)dt

+

∫ 2−2h−√4h2−4h+2

√3−6h+4h2+2(h−1)

√4h2−4h+2

K2(t)dt+

∫ 1−2h

2−2h−√4h2−4h+2

K3(t)dt

+

∫ 1

1−2h−K3(t)dt = η(h).

The inequality (ii) follows considering the following estimation of remainder term

|R[f ]| ≤ ∥f ′′∥2[∫ 1

−1K(t)2dt

]1/2.

We have∫ 1

−1K(t)2dt =

∫ −(2−2h−√4h2−4h+2)

−1K1(t)

2dt+

∫ 2−2h−√4h2−4h+2

−(2−2h−√4h2−4h+2)

K2(t)2dt

+

∫ 1

2−2h−√4h2−4h+2

K3(t)2dt = ν(h)2.

Theorem 5 Let f : [−1, 1] → R be a function that f ′ ∈ L1[−1, 1]. If there exists areal number γ, such that γ ≤ f ′(t), t ∈ [−1, 1], then

|R[f ]| ≤ 2h(S − γ),

and if there exist a real number Γ such that f ′(t) ≤ Γ, t ∈ [−1, 1], then

|R[f ]| ≤ 2h(Γ− S),

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Error bounds for a class of quadrature formulas 145

where S =f(1)− f(−1)

2and h ∈

[0,

1

2

]. If there exist real numbers γ,Γ such that

γ ≤ f ′(t) ≤ Γ, t ∈ [−1, 1], then

|R[f ]| ≤ 1

2φ(h)(Γ− γ),

where φ(h) are defined as

φ(h) = 4h2 − 6h+ 3 + 2(h− 1)√

4h2 − 4h+ 2.

Proof. Let us define

p(t) =

t+ 1− h, t ∈ [−1,−x],t, t ∈ (−x, x),t− 1 + h, t ∈ [x, 1],

where x = 2− 2h−√4h2 − 4h+ 2.

Since

∫ 1

−1p(t)dt = 0 and

∫ 1

−1p(t)f ′(t)dt = −R[f ], it follows

|R[f ]| =∣∣∣∣∫ 1

−1p(t)f ′(t)dt

∣∣∣∣ = ∣∣∣∣∫ 1

−1

(f ′(t)− γ

)p(t)dt

∣∣∣∣ ≤ ∥p∥∞ · ∥f ′ − γ∥1 = 2h(S − γ)

and

|R[f ]| =∣∣∣∣∫ 1

−1p(t)f ′(t)dt

∣∣∣∣ = ∣∣∣∣∫ 1

−1

(Γ− f ′(t)

)p(t)dt

∣∣∣∣ ≤ ∥p∥∞ · ∥Γ− f ′∥1 = 2h(Γ− S).

Since

∣∣∣∣f ′(t)− γ + Γ

2

∣∣∣∣ ≤ Γ− γ

2, we obtain

|R[f ]| =∣∣∣∣∫ 1

−1p(t)

(f ′(t)− γ + Γ

2

)dt

∣∣∣∣ ≤ Γ− γ

2∥p∥1 = φ(h)

Γ− γ

2.

3 The corrected quadrature formulas

In the last years many authors have considered so called corrected quadrature rules([2]-[6],[11]). In this section we will construct the corrected quadrature formulasof the optimal quadrature formula (10) and we show that the estimations usingdifferent norms are better in the corrected formula than in the original one.

Let

A :=

∫ 1

−1K(t)dt = 4h3 − 10h2 +

17

2h− 17

6+ 2(h− 1)2

√4h2 − 4h+ 2.

Page 146: 2014 Volume 22 No. 1

146 F. Sofonea, A. M. Acu, A. Rafiq, D. Barbosu

The corrected quadrature formula of (10) is defined bellow:

∫ 1

−1f(x)dx = hf(−1) + (1− h)f

(2− 2h−

√4h2 − 4h+ 2

)(12)

+ (1− h)f(−2 + 2h+

√4h2 − 4h+ 2

)+ hf(1) +A

[f ′(1)− f ′(−1)

]+ R[f ],

where

R[f ] =

∫ 1

−1K(t)f ′′(t)dt and K(t) = K(t)−A.

Theorem 6 For the remainder term of the quadrature formula (12) the followingestimations can be established ∣∣∣R[f ]

∣∣∣ ≤ Ω(h) · ∥f ′′∥1,

where

Ω(h) = −4h3 + 12h2 − 23

2h+

13

3+(−2h2 + 5h− 3

)√4h2 − 4h+ 2.

Remark 2 From the below figure we remark that there are values of h such that theestimation in the corrected formula is better than in the original one.

0 0.1 0.2 0.3 0.4 0.50.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

Ψ(h)

Ω(h)

Figure 1.

Theorem 7 For the remainder term of the quadrature formula (12) the followingestimation can be established ∣∣∣R[f ]

∣∣∣ ≤ ν∗(h) · ∥f ′′∥2,

where

ν∗(h) = −64h6 +896

3h5 − 592h4 + 644h3 − 2455

6h2 +

871

6h− 1018

45

+√

4h2 − 4h+ 2

(−32h5 +

400

3h4 − 676

3h3 +

584

3h2 − 260

3h+ 16

).

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Error bounds for a class of quadrature formulas 147

Remark 3 In Figure 2 are represented the coefficients ν, ν∗ which appear in esti-mations of the remainder term in L2-norm for the original and corrected quadratureformulas. The estimation in the corrected formula is better than in the original one.

0 0.1 0.2 0.3 0.4 0.50

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

ν(h)

ν*(h)

Figure 2.

Let f, g : [a, b] → R be integrable functions on [a, b]. The functional

T (f, g) :=1

b− a

∫ b

af(t)g(t)dt− 1

b− a

∫ b

af(t)dt · 1

b− a

∫ b

ag(t)dt,

is well known in the literature as the Cebysev functional and the inequality

|T (f, g)| ≤√T (f, f) ·

√T (g, g)

holds. Denote by σ(f, a, b) =√

(b− a)T (f, f).

Theorem 8 Let f : [−1, 1] → R be a twice differentiable function such that f ′′ isintegrable. Then, ∣∣∣R[f ]

∣∣∣ ≤ ν∗(h)σ(f ′′,−1, 1).

Proof. We have

R[f ] =

∫ 1

−1K(t)f ′′(t)dt =

∫ 1

−1

[K(t)− 1

2

∫ 1

−1K(t)dt

]f ′′(t)dt

=

∫ 1

−1K(t)f ′′(t)dt− 1

2

∫ 1

−1K(t)dt

∫ 1

−1f ′′(t)dt = 2T (K, f ′′).

Therefore ∣∣∣R[f ]∣∣∣ ≤√2T (K,K) ·

√2T (f ′′, f ′′) = ν∗(h)σ(f ′′,−1, 1).

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148 F. Sofonea, A. M. Acu, A. Rafiq, D. Barbosu

References

[1] A.M. Acu, F. Sofonea, A class of optimal quadrature formula, Acta UniversitatisApulensis, Special Issue, 2011, 481-489.

[2] P. Cerone, S. S. Dragomir, Midpoint-type rules from an inequalities point ofview, Handbook of Analytic-Computational Methods in Applied Mathematics,Editor: G. Anastassiou, CRC Press, New York, (2000), 135–200.

[3] P. Cerone, S. S. Dragomir, Trapezoidal-type rules from an inequalities point ofview, Handbook of Analytic-Computational Methods in Applied Mathematics,Editor: G. Anastassiou, CRC Press, New York, (2000), 65–134.

[4] P. Cerone, Three point rules in numerical integration, J. Non-linear Analysis,47, (2001), 2341-2352.

[5] S. S. Dragomir, R. P. Agarwal, P. Cerone, On Simpson’ s inequality and appli-cations, J. Inequal. Appl., 5 (2000), 533–579.

[6] I. Franjic, J. Pecaric, On corrected Bullen-Simpson’ s 3/8 inequality, TamkangJournal of Mathematics, 37(2), (2006), 135–148.

[7] N. Ujevic, Error inequalities for a quadrature formula and applications,Computers & Mathematics with Applications, vol. 48, no. 10-11, 2004,1531–1540.

[8] N. Ujevic, Error inequalities for a quadrature formula of open type, RevistaColombiana de Matematicas, vol. 37, no. 2, 93–105, 2003.

[9] N. Ujevic, Error inequalities for an optimal quadrature formula, Journal ofApplied Mathematics and Computing, vol. 24, no. 1-2, 65–79, 2007.

[10] N. Ujevic, Error inequalities for a quadrature formula of open type, RevistaColombiana de Matematicas, Volumen 37, 2003, 93-105.

[11] N. Ujevic, A. J. Roberts, A corrected quadrature formula and applications,ANZIAM J. 45 (2004), 41–56.

[12] F. Zafar, N.A.Mir, Some generalized error inequalities and applications, Journalof Inequalities and Applications, Volume 2008, Article ID 845934, 15 pages.

Ana Maria Acu, Daniel Florin SofoneaLucian Blaga University of SibiuFaculty of SciencesDepartment of Mathematics and InformaticsStr. Dr. I. Ratiu, No.5-7, 550012 Sibiu, Romaniae-mail: [email protected], [email protected]

Page 149: 2014 Volume 22 No. 1

Error bounds for a class of quadrature formulas 149

Arif RafiqLahore Leads UniversityDepartment of MathematicsLahore, Pakistane-mail: [email protected]

Dan BarbosuNorth University Center at Baia MareTechnical University of Cluj-NapocaFaculty of SciencesDepartment of Mathematics and Computer ScienceStr. Victoriei 76, 430122 Baia Mare, Romaniae-mail: [email protected]