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

Abstract 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 : | 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 U R / q if and only if f is linear.