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

Abstract

Let $G$ be a domain in the complex plane $\mathbb{C}$ bounded by a rectifiable Jordan curve $\Gamma$, let $z_0\in G$ and let $\varphi_0$ be the Riemann conformal map of $G$ onto $\mathbb{D}_r:=\{ w\in\mathbb{C}\,:\, |w|<r \}$, normalized by $\varphi_0(z_0)=0$, $\varphi_0'(z_0)=1$. In this work the simultaneous approximations of $\varphi_0$ and its derivatives by Bieberbach polynomials are investigated. The approximation rate in dependence of the smoothness parameters of the considered domains is estimated.