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 UR := {z: |z| 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 UR/q if and only if f is linear. © 2007 Elsevier Inc. All rights reserved.