Simultaneous approximation of the Riemann conformal map and its derivatives by Bieberbach polynomials

Abstract

Let G be a domain in the complex plane C bounded by a rectifiable Jordan curve Gamma, let z(0) is an element of G and let phi(0) be the Riemann conformal map of G onto D-r= {w is an element of C:vertical bar w vertical bar<r}, normalized by phi(0) (z(0)) = 0, phi(0) (z(0)) = 1. In this work the simultaneous approximations of phi(0) and its derivatives by Bieberbach polynomials are investigated. The approximation rate in dependence of the smoothness parameters of the considered domains is estimated.