The saturation of convergence on the interval [0,1] for the q-Bernstein polynomials in the case q>1

In the note, we consider saturation of convergence on the interval [0,1] for the q-Bernstein polynomials of a continuous function f for arbitrary fixed q>1. We show that the rate of uniform convergence on [0,1] is o(q−n) if and only if f is linear. The result is sharp in the following sense: it ceases to be true if we replace “o” by “O”.