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

AbstractIn 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|<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 UR/q if and only if f is linear