Results Math Online First c  2016 Springer International Publishing DOI 10.1007/s00025-016-0639-3 Results in Mathematics Bezier variant of the Bernstein–Durrmeyer type operators Tuncer Acar, P. N. Agrawal, and Trapti Neer Abstract. In the present paper, we introduce the Bezier-variant of Dur- rmeyer modification of the Bernstein operators based on a function τ , which is infinite times continuously differentiable and strictly increasing function on [0, 1] such that τ (0) = 0 and τ (1) = 1. We give the rate of approximation of these operators in terms of usual modulus of continu- ity and K-functional. Next, we establish the quantitative Voronovskaja type theorem. In the last section we obtain the rate of convergence for functions having derivative of bounded variation. Mathematics Subject Classification. 41A25, 41A35. Keywords. Bezier operators, K-functional, Modulus of continuity, Functions of bounded variation. 1. Introduction In 1912, Bernstein [6] defined a sequence of positive linear operators for f ∈ C[0, 1], as B n (f ; x)= n  k=0 f  k n  n k  x k (1 − x) n−k , x ∈ [0, 1] which preserves linear functions. To make convergence faster, King [12] intro- duced a modification of these operators as ((B n f ) ◦ r n )(x)= n  k=0 f  k n  n k  (r n (x)) k (1 − r n (x)) n−k , which depends on a sequence r n (x) of continuous functions on [0, 1] with 0 ≤ r n (x) ≤ 1, for each x ∈ [0, 1] and considered a particular case for the sequence