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In this paper, in order to make the convergence faster to a function being approximated, we introduce a kind of complex modified q-Durrmeyer type operators which can reproduce constant and linear functions. We study the approximation properties of these operators. We obtain the order of simultaneous approximation and a Voronovskaja-type result with a quantitative estimate for these complex modified q-Durrmeyer type operators attached to analytic functions on compact disks. More important, our results show the overconvergence phenomenon for these complex operators. MSC: 30E10, 41A25. © 2012 Ren and Zeng; licensee Springer.

Approximation by a kind of complex modified q-Durrmeyer type operators in compact disks