-Parametric Bleimann Butzer and Hahn Operators
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Research Article q-Parametric Bleimann Butzer and Hahn Operators N. I. Mahmudov and P. Sabancıgil Eastern Mediterranean University, Gazimagusa, Turkish Republic of Northern Cyprus, Mersin 10, Turkey Correspondence should be addressed to N. I. Mahmudov, [email protected] Received 4 June 2008; Accepted 20 August 2008 Recommended by Vijay Gupta We introduce a new q-parametric generalization of Bleimann, Butzer, and Hahn operators in ∗ 0, ∞. We study some properties of q-BBH operators and establish the rate of convergence C1x for q-BBH operators. We discuss Voronovskaja-type theorem and saturation of convergence for qBBH operators for arbitrary fixed 0 < q < 1. We give explicit formulas of Voronovskaja-type for the q-BBH operators for 0 < q < 1. Also, we study convergence of the derivative of q-BBH operators. Copyright q 2008 N. I. Mahmudov and P. Sabancıgil. 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.
1. Introduction q-Bernstein polynomials Bn,q fx :
n−k−1 n k n k f 1 − qs x x k n s0 k0
1.1
were introduced by Phillips in 1. q-Bernstein polynomials form an area of an intensive research in the approximation theory, see survey paper 2 and references therein. Nowadays, there are new studies on the q-parametric operators. Two parametric generalizations of qBernstein polynomials have been considered by Lewanowicz and Wo´zny cf. 3, an analog of the Bernstein-Durrmeyer operator and Bernstein-Chlodowsky operator related to the qBernstein basis has been studied by Derriennic 4, Gupta 5 and Karsli and Gupta 6, respectively, a q-version of the Szasz-Mirakjan operator has been investigated by Aral and Gupta in 7. Also, some results on q-parametric Meyer-Konig and Zeller operators can be ¨ found in 8–11. In 12, Bleimann et al. introduced the following operators: Hn fx
n 1 k n f xk , n k n−k1 1 x k0
x > 0, n ∈ N.
1.2
2
Journal of Inequalities and Applications
There are several studies related to approximation properties of Bleimann, Butzer, and Hahn operators or, briefly, BBH, see, for example, 12–18. Recently, Aral and Dogru ˘ 19 introduced a q-analog of Bleimann, Butzer, and Hahn operators and they have established some approximation properties of their q-Bleimann, Butzer, and Hahn operators in the subspace of CB 0, ∞. Also, they showed that these operators are more flexible than classical BBH operators, that is, depending on the selection of q, rate of convergence of the q-BBH operators is better than the classical one. Voronovskaja-type asymptotic estimate and the monotonicity properties for q-BBH operators are studied in 20. In this paper, we propose a different q-analog of the Bleimann, Butzer, and Hahn ∗ 0, ∞. We use the connection between classical BBH and Bernstein operators in C1x operators suggested in 16 to define new q-BBH operators as follows: Hn,q
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