Acta Mathematica Scientia 2014,34B(4):1157–1165 http://actams.wipm.ac.cn APPROXIMATION BY COMPLEX SZ ´ ASZ-DURRMEYER OPERATORS IN COMPACT DISKS ∗ Sorin G. GAL Department of Mathematics and Computer Science, University of Oradea Str. Universitatii No. 1, 410087 Oradea, Romania E-mail : galso@uoradea.ro Vijay GUPTA † Department of Mathematics, Netaji Subhas Institute of Technology Sector 3 Dwarka, New Delhi-110078, India E-mail : vijaygupta2001@hotmail.com Abstract In the present paper, we deal with the complex Sz´asz-Durrmeyer operators and study Voronovskaja type results with quantitative estimates for these operators attached to analytic functions of exponential growth on compact disks. Also, the exact order of approximation is found. Key words complex Sz´asz-Durrmeyer operators; Voronovksaja type result; exact order of approximation in compact disks; simultaneous approximation 2010 MR Subject Classification 41A25; 41A28; 30E10 1 Introduction Concerning the convergence of the Bernstein polynomials in the complex plane, Bernstein [10] proved that if f : G → C is analytic in the open set G ⊂ C, with D 1 ⊂ G (with D 1 = {z ∈ C : |z | < 1}), then the complex Bernstein polynomials B n (f )(z )= n ∑ k=0 ( n k ) z k (1 − z ) n−k f (k/n), uniformly converges to f in D 1 . Voronovskaja-type results with quantitative estimates for the complex Bernstein, complex q-Bernstein, complex Baskakov, complex Sz´ asz (more precisely Favard-Sz´ asz-Mirakjan), com- plex Bernstein-Kantorovich, complex Bal´ azs-Szabados and complex Stancu-Kantorovich op- erators attached to analytic functions on compact disks and the exact order of simultaneous approximation by these complex operators were collected by the recent book Gal [1]. Recall that for f ∈ C[0, ∞), x ∈ [0, ∞) and n ∈ N, the classical Sz´ asz operator of real variable is defined as S n (f )(x)= ∞  ν=0 s n,ν (x)f  ν n  , * Received March 25, 2013; revised December 9, 2013. † Corresponding author: Vijay GUPTA.