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Since for q > 1, q-Bernstein polynomials are not positive linear operators on C(0,1), the investigation of their convergence properties turns out to be much more difficult than that in the case 0 1, the q-Bernstein polynomials approximate the entire functions and, in particular, polynomials uniformly on any compact set in C. In this paper, the possibility of the approximation for the function (z + a) α , a 0, with a non-integer α > −1 is studied. It is proved that for a > 0, the function is uniformly approximated on any compact set in {z : |z| a}, the uniform approximation is impossible. In the case a = 0, the results of the paper reveal the following interesting phenomenon: the power function z α , α > 0, is approximated by its q-Bernstein polynomials either on any (when α ∈ N) or no (when α / ∈ N) Jordan arc in C.

The approximation of power function by the q-Bernstein polynomials in the case qϡ1