DOI: 10.2478/s12175-013-0152-z Math. Slovaca 63 (2013), No. 5, 1025–1036 APPROXIMATION BY COMPLEX SUMMATION-INTEGRAL TYPE OPERATOR IN COMPACT DISKS Vijay Gupta — Rani Yadav (Communicated by J´ an Bors´ ık ) ABSTRACT. In the present paper we estimate a Voronovskaja type quantitative estimate for a certain type of complex Durrmeyer polynomials, which is different from those studied previously in the literature. Such estimation is in terms of analytic functions in the compact disks. In this way, we present the evidence of overconvergence phenomenon for this type of Durrmeyer polynomials, namely the extensions of approximation properties (with quantitative estimates) from real intervals to compact disks in the complex plane. In the end, we mention certain applications. c 2013 Mathematical Institute Slovak Academy of Sciences 1. Inroduction If f : G → C is an analytic function in the open set G ⊂ C, with D 1 ⊂ G (where D 1 = {z ∈ C : |z | < 1}), then S. N. Bernstein proved that the complex Bernstein polynomials converges uniformly to f in D 1 (see e.g., Lorentz [8: p. 88]). Sorin G Gal has done commendable work in this direction and he estimated upper quantitative estimates for the uniform convergence for the first time. (see e.g. [3: p. 264]). Also exact quantitative estimates for different operators were established in his recent papers see e.g. [2], [4], [6] and [5] etc. In the recent years and for the real variable case, Abel-Gupta-Mohapatra [1] studied the rate of convergence and established asymptotic expansion of certain Bernstein-Durrmeyer type operators, which are discretely defined at f (0). The aim of the present article is to extend the studies on such operators. Let R> 1 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 30E10; Secondary 41A25. K e y w o r d s: complex Durrmeyer-type operators, uniform convergence, analytic function, Voronovskaja-type result.