The saturation of convergence on the interval for the q-Bernstein polynomials in the case

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”.