Saturation of convergence for q-Bernstein polynomials in the case q ≥ 1

In the note, we discuss Voronovskaya type theorem and saturation of convergence for q-Bernstein polynomials for a function analytic in the disc U-R := {Z: vertical bar z vertical bar < R} (R > q) for arbitrary fixed q >= 1. We give explicit formulas of Voronovskaya type for the q-Bernstein polynomials for q > 1. We show that the rate of convergence for the q-Bernstein polynomials is o(q(-n)) (q > 1) for infinite number of points having an accumulation point on U-R/q if and only if f is linear. (C) 2007 Elsevier Inc. All rights reserved.