Approximation by q-Durrmeyer type polynomials in compact disks in the case q > 1

Abstract

Recently, Agarwal and Gupta (2012) [1] studied some approximation properties of the complex q-Durrmeyer type operators in the case 0 < q < 1. In this paper this study is extended to the case q > 1. More precisely, approximation properties of the newly defined generalization of this operators in the case q > 1 are studied. Quantitative estimates of the convergence, the Voronovskaja type theorem and saturation of convergence for complex q-Durrmeyer type polynomials attached to analytic functions in compact disks are given. In particular, it is proved that for functions analytic in {z is an element of C :vertical bar z vertical bar < R}, R > q, the rate of approximation by the q-Durrmeyer type polynomials (q > 1) is of order q-n versus 1/n for the classical (q = 1) Durrmeyer type polynomials. Explicit formulas of Voronovskaya type for the q-Durrmeyer type operators for q > 1 are also given. This paper represents an answer to the open problem initiated by Gal (2013) [6]. (C) 2014 Elsevier Inc. All rights reserved.