Abstract
Recently, Gupta and Wang introduced certain q-Durrmeyer type operators of real variable x ∈ [0, 1] and studied some approximation results in the case of real variables. Here we extend this study to the complex variable for analytic functions in compact disks. We establish the quantitative Voronovskaja type estimate. In this way, we put in evidence the over convergence 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. Some of these results for q = 1 were recently established in Gupta-Yadav. Mathematical subject classification (2000): 30E10; 41A25.
Highlights
In the recent years applications of q-calculus in the area of approximation theory and number theory is an active area of research
Several researchers have proposed the q analogue of exponential, Kantorovich and Durrmeyer type operators
The main contributions for the complex operators are due to Gal; several important results have been complied in his recent monograph [4]
Summary
In the recent years applications of q-calculus in the area of approximation theory and number theory is an active area of research. The aim of the present article is to extend approximation results for such q-Durrmeyer operators to the complex case. Gal and Gupta [5,6,7] have studied some other complex Durrmeyer type operators, which are different from the operators considered in the present article. We shall study approximation results for the complex q-Durrmeyer operators (introduced and studied in the case of real variable by Gupta-Wang [3]), defined by n
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