Numer Algor (2012) 59:119–129 DOI 10.1007/s11075-011-9479-0 ORIGINAL PAPER New estimates in Voronovskaja’s theorem Gancho Tachev Received: 10 March 2011 / Accepted: 27 May 2011 / Published online: 14 July 2011 © Springer Science+Business Media, LLC 2011 Abstract In the present article we establish pointwise variant of E. V. Voronovskaja’s 1932 result, concerning the degree of approximation of Bernstein operator, applied to functions f ∈ C 3 [0, 1]. Keywords Bernstein polynomials · Voronovskaja-type theorem · Moduli of continuity · Degree of approximation 1 Introduction For every function f ∈ C[0, 1] the Bernstein polynomial operator is given by B n ( f ; x) = n  k=0 f ( k n ) ·  n k  x k (1 − x) n−k , x ∈[0, 1] For this operator the theorem of Voronovskaja was first proved in [20] and is given in the book of DeVore and Lorentz [2] as follows: Theorem A If f is bounded on [0, 1], dif ferentiable in some neighborhood of x and has second derivative f ′′ (x) for some x ∈[0, 1], then lim n→∞ n ·[ B n ( f, x) − f (x)]= x(1 − x) 2 · f ′′ (x). If f ∈ C 2 [0, 1], the convergence is uniform. G. Tachev (B ) University of Architecture, 1 Hr.Smirnenski Blvd., Sofia, 1046, Bulgaria e-mail: gtt_fte@uacg.bg