Approximation by a complex q-durrmeyer type operator

Abstract

Very recently, for 0 < q < 1 Govil and Gupta [10] introduced a certain q-Durrmeyer type operators of real variable \({x \in [0,1]}\) and established some approximation properties. In the present paper, for these q-Durrmeyer operators, 0 < q < 1, but of complex variable z attached to analytic functions in compact disks, we study the exact order of simultaneous approximation and a Voronovskaja kind result with quantitative estimate. In this way, we put in evidence the overconvergence phenomenon for these q-Durrmeyer polynomials, namely the extensions of approximation properties (with quantitative estimates) from the real interval [0, 1] to compact disks in the complex plane. For q = 1 the results were recently proved in Gal-Gupta [8].