Complex operators generated by q-Bernstein polynomials, q >= 1

By using a univalent and analytic function tau in a suitable open disk centered in origin, we attach to analytic functions f, the complex Bernsteintype operators of the form B-n,q(tau) (f) = B-n,B-q (f o T-1)o T, where B-n,B-q denote the classical complex q- Bernstein polynomials, q >= 1. The new complex operators satisfy the same quantitative estimates as B-n,B-q. As applications, for two concrete choices of T, we construct complex rational functions and complex trigonometric polynomials which approximate f with a geometric rate.