Abstract and Applied Analysis
Volume 2015 (2015), Article ID 480925, 7 pages
http://dx.doi.org/10.1155/2015/480925

Research Article

On -Bleimann, Butzer, and Hahn-Type Operators

Dilek Söylemez

Ankara University, Elmadag Vocational School, Department of Computer Programming, 06780 Ankara, Turkey

Received 16 October 2015; Revised 14 December 2015; Accepted 15 December 2015

Academic Editor: Simeon Reich

Copyright © 2015 Dilek Söylemez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Sequence of -Bleimann, Butzer, and Hahn operators which is based on a continuously differentiable function on , with , , has been considered. Uniform approximation by such a sequence has been studied and degree of approximation by the operators has been obtained. Moreover, shape preserving properties of the sequence of operators have been investigated.

1. Introduction

In 1987, Lupaş [1] introduced the first -analogue of Bernstein operators and investigated its approximation and shape preserving properties. Another -generalization of the classical Bernstein polynomials is due to Phillips [2]. After that many generalizations of positive linear operators based on -integers were introduced and studied by several authors. Some are in [3–13].

Bleimann et al. [14] proposed a sequence of positive linear operators defined by

for , where denote the space of all continuous and real valued functions defined on Also the authors proved that as pointwise on for any , where denote the space of all bounded functions from . It is well known that

defines a norm on . Moreover, they showed that the convergence is uniform on each compact subset of . In [15], using test functions for , Gadjiev and Çakar stated a Korovkin-type theorem for the uniform approximation of functions belonging to some suitable function spaces by some linear positive operators. As an application of this result, they proved uniform approximation by Bleimann, Butzer, and Hahn operators. Further results concerning such a sequence of operators and its generalizations may be found in [16–19].

Now, we recall some notations from -analysis [20, 21].

The -integer and the -factorial are defined by

, and

respectively, where .For integers the -binomial coefficient is defined as

Moreover, Euler identity is given by

Aral and Doğru [22] constructed the -Bleimann, Butzer, and Hahn operators as

where

and is defined on the semiaxis The authors studied Korovkin-type approximation properties by using the test functions for . Moreover, they obtained rate of convergence of the operators and proved that rate of the -Bleimann, Butzer, and Hahn operators is better than the classical one. A generalization of the -Bleimann, Butzer, and Hahn operators was introduced by Agratini and Nowak in [23]. In this paper, the authors gave representation of the operators in terms of -differences and investigated some approximation properties.

A Voronovskaja-type result and monotonicity properties of these operators are investigated in [24].

In [25], the authors introduced a new generalization of Bernstein polynomials denoted by and defined as

where is the th Bernstein polynomial, , , and is a function that is continuously differentiable of infinite order on such that , , and for . Also, the authors studied some shape preserving and convergence properties concerning the generalized Bernstein operators .

In [26], Aral et al. constructed sequences of Szasz-Mirakyan operators which are based on a function . They studied weighted approximation properties and Voronovskaja-type results for these operators. They also showed that the sequence of the generalized Szász-Mirakyan operators is monotonically nonincreasing under the -convexity of the original function. A similar generalization for Bleimann, Butzer, and Hahn operators is studied by Söylemez [27]. Also the class was defined, a Korovkin-type theorem was given for the functions in this class, and uniform convergence of the generalized Bleimann, Butzer, and Hahn operators was obtained [27]. Moreover, the monotonicity properties of the operators were investigated.

Now we recall the definition of that is a subspace of [27].

Let be a general modulus of continuity, satisfying the following properties:

(a)is continuous, nonnegative, and increasing function on ,(b),(c)

The space of all real valued functions defined on satisfying

for all is denoted by

It is clear from condition (b) that we have

and one can get from the condition (a) that for any

where denotes the greatest integer that is not greater than .

Now we define a new generalization of -Bleimann, Butzer, and Hahn operators for by

where

and is a continuously differentiable function on such that

An example of such a function is given in [26]. Note that, in the setting of the operators (13), we have

where the operators are defined by (7). If , then Obviously, we have

In this study, we consider a generalization of -Bleimann, Butzer, and Hahn operators in the sense of [26], we investigate uniform convergence of to on for , and we obtain the degree of approximation. Moreover, we study shape preserving properties under -convexity of the function. Our results show that the new operators are sensitive to the rate of convergence to , depending on the selection of . For the particular case , the previous results for -Bleimann Butzer and Hahn operators are obtained.

In order to ensure that the convergence properties holds, the author will assume is a sequence such that as for , as in [22].

Definition 1. Let be distinct points in the domain of . Denote

where remains fixed and takes all values from to , excluding .

Definition 2. A continuous, real valued function is said to be convex in , if

for every and for every nonnegative number of such that .

In [25] Cárdenas-Morales et al. introduced the following definition of -convexity of a continuous function.

Definition 3. A continuous, real valued function is said to be -convex in , if is convex in the sense of Definition 2.

2. Approximation Properties

In this section we deal with the promised approximation properties of the sequence of -Bleimann, Butzer, and Hahn operators. In [27], the following Korovkin-type theorem was given.

Theorem 4. Let be a sequence of linear positive operators from to If

is satisfied for , then for one has

Now we are ready to give the following theorem.

Theorem 5. Suppose that , , and let as If is the operator defined by (13), then for any one has

Proof. According to Theorem 4 we will show that (20) holds for . Obviously, from (17) we easily obtain that

Therefore, the conditions (20) are satisfied. By Theorem 4, the proof is completed.

Theorem 6. Let , , and let as Then one has

for all . Here,

Proof. Firstly, from (12) and (10) we can write

and considering (17), we get

Equations (27) and (26) together imply that

Using the Cauchy-Schwarz inequality, we obtain

By choosing , we get

where

which concludes the proof.

3. Shape Preserving Properties

Theorem 7. Let be a -convex function that is nonincreasing on ; then one has

for .

Proof. From (13), one can write

Moreover, we have the following equalities that are proved in Lemma 3.1 of [24]:

which imply

From and by hypothesis, we have

Since is -convex, by using Lemma  3.2 in [24] we obtain

for and This proves the theorem.

Theorem 8. Suppose that , and is linear. Then one has

Proof. From (17), we have

Using the equality , we now get

On the other hand we can write

which implies

Now taking (42) in (40), we have

Thus the proof is completed.

Now, from Theorem 8, we have the following corollary immediately.

Corollary 9. Let be a -convex, nonincreasing function on and is linear. Then one has

for any ,

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

References

A. Lupaş, “A q-analogue of the Bernstein operators,” in Seminar on Numerical and Statistical Calculus, vol. 9, Technical University of Cluj-Napoca, Cluj-Napoca, Romania, 1987.

G. M. Phillips, “Bernstein polynomials based on the q-integers,” Annals of Numerical Mathematics, vol. 4, pp. 511–518, 1997.

A. Il'inskii and S. Ostrovska, “Convergence of generalized Bernstein polynomials,” Journal of Approximation Theory, vol. 116, no. 1, pp. 100–112, 2002.

H. Oruç and N. Tuncer, “On the convergence and iterates of -Bernstein polynomials,” Journal of Approximation Theory, vol. 117, no. 2, pp. 301–313, 2002.

S. Ostrovska, “q-Bernstein polynomials and their iterates,” Journal of Approximation Theory, vol. 123, no. 2, pp. 232–255, 2003.

A. Aral and V. Gupta, “On q-Baskakov type operators,” Demonstratio Mathematica, vol. 42, no. 1, 2009.

A. Aral and V. Gupta, “q-derivative and applications to the q-Szász Mirakyan operators,” Calcolo, vol. 43, pp. 151–170, 2006.

M. Mursaleen and K. J. Ansari, “Approximation of q-Stancu-Beta operators which preserve x ²,” Bulletin of the Malaysian Mathematical Sciences Society, 2015.

M. Mursaleen, F. Khan, A. Khan, and A. Kiliçman, “Some approximation results for generalized Kantorovich-type operators,” Journal of Inequalities and Applications, vol. 2013, article 585, 2013.

M. Mursaleen, F. Khan, and A. Khan, “Approximation properties for King's type modified q-Bernstein-Kantorovich operators,” Mathematical Methods in the Applied Sciences, 2015.

M. Mursaleen, F. Khan, A. Khan, and A. Kiliçman, “Some approximation properties of Bivariate q-stancu-β operators,” Journal of Function Spaces, vol. 2014, Article ID 270673, 7 pages, 2014.

M. Mursaleen and A. Khan, “Generalized q-Bernstein-Schurer operators and some approximation theorems,” Journal of Function Spaces and Applications, vol. 2013, Article ID 719834, 7 pages, 2013.

M. Mursaleen and A. Khan, “Statistical approximation properties of modified q-Stancu-β operators,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 36, no. 3, pp. 683–690, 2013.

G. Bleimann, P. L. Butzer, and L. Hahn, “A Bernstein-type operator approximating continuous functions on the semi-axis,” Indagationes Mathematicae (Proceedings), vol. 83, no. 3, pp. 255–262, 1980.

A. D. Gadjiev and Ö. Çakar, “On uniform approximation by Bleimann, Butzer and Hahn operators on all positive semi-axis,” Transactions of AS Azerbaijan. Series of Physical Technical and Mathematical Sciences, vol. 19, no. 5, pp. 21–26, 1999.

U. Abel and M. Ivan, “Some identities for the operator of Bleimann, Butzer and Hahn involving divided differences,” Calcolo, vol. 36, no. 3, pp. 143–160, 1999.

R. A. Khan, “A note on a Bernstein-type operator of Bleimann, Butzer, and Hahn,” Journal of Approximation Theory, vol. 53, no. 3, pp. 295–303, 1988.

J. A. Adell, J. De la Cal, and M. San Miguel, “Inverse beta and generalized Bleimann-Butzer-Hahn operators,” Journal of Approximation Theory, vol. 76, no. 1, pp. 54–64, 1994.

B. Della Vecciha, Some Properties of a Rational Operator of Bernstein-Type, Istituto per Applicazioni della Matematica-C.N.R. Via P. Castellino, Napoli, Italy, 1991.

T. Ernst, “The history of q-calculus and a new method,” U.U.U.D.M Report 16, Department of Mathematics, Upsala University, 2000.

G. M. Phillips, Interpolation and Approximation by Polynomials, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 14, Springer, New York, NY, USA, 2003.

A. Aral and O. Doğru, “Bleimann, butzer and hahn operators on the q-Integers,” Journal of Inequalities and Applications, vol. 2007, Article ID 79410, 12 pages, 2007.

O. Agratini and G. Nowak, “On the generalizations of Bleimann, Butzer and Hahn operators based on q-integers,” Mathematical and Computer Modelling, vol. 53, pp. 699–706, 2011.

O. Doğru and V. Gupta, “Monotonicity and the asymptotic estimate of Bleimann, Butzer and Hahn operators based on q-integers,” Georgian Mathematical Journal, vol. 12, no. 3, pp. 415–422, 2005.

D. Cárdenas-Morales, P. Garrancho, and I. Raşa, “Bernstein-type operators which preserve polynomials,” Computers & Mathematics with Applications, vol. 62, no. 1, pp. 158–163, 2011.

A. Aral, D. Inoan, and I. Raşa, “On the generalized Szász-Mirakyan operators,” Results in Mathematics, vol. 65, no. 3-4, pp. 441–452, 2014.

D. Söylemez, “Some properties of the generalized Bleimann, Butzer and Hahn operators,” Communications Faculty of Science University of Ankara Series A1, vol. 64, no. 2, pp. 55–62, 2015.