Uniform convergence of the Bieberbach polynomials in closed smooth domains of bounded boundary rotation

Abstract

Let G be a Jordan smooth domain of bounded boundary rotation, let z(0) epsilon G, and let w = phi(0) (z) be the conformal mapping of G onto D (0, r(0)) := {w : \w\ < r(0)) with the normalization phi(0) (z(0)) = 0, phi(0)' (z(0)) = 1. Let also pi(n) (z), n = 1, 2,..., be the Bieberbach polynomials for the pair (G, z(0)). We investigate the uniform convergence of these polynomials on 6 and prove the 0 estimate \\ phi(0) - pi(n)\\ ((G) over bar) (.)= (zepsilon (G) over bar)max \phi(0) (z) - pi(n9)(z)\ less than or equal to (c) under bar /n(1 - epsilon') for some constant e = (epsilon) independent of n.