How Analytic Properties of Functions Influence Their Images Under the Limit q-Stancu Operator

Abstract

In the study of various q-versions of the Bernstein polynomials, a significant attention is paid to their limit operators. The present work focuses on the impact of the limit q-Stancu operator Sq infinity,alpha on the analytic properties of functions when 0 < q < 1 and alpha > 0. It is shown that for every f is an element of C[0, 1], the function S-q,(alpha infinity)fadmits an analytic continuation into the disk {z : z+alpha/(1-q) < 1+ alpha/(1-q)}. In addition, it is proved that the more derivatives f has at x = 1, the wider this disk becomes. Further, if f is infinitely differentiable at x = 1, then the function S-q,(alpha infinity)fis entire. Finally, some growth estimates for (S-q,(alpha infinity)f)(z) are obtained.