Convergence of Bieberbach polynomials inside domains of the complex plane

Abstract

Let $G\subset C $ be a finite Jordan domain, $z_{0}\in G;$ $B\Subset G$ be an arbitrary closed disk with $z_{0}\in B,$ and $w=\varphi (z,z_{0})$ be the conformal mapping of $G$ onto a disk $\{w:\left| w\right| <r\}$ normalized by $\varphi (z_{0},z_{0})=0$, $\varphi ^{\prime }(z_{0},z_{0})=1$ . It is well known that the Bieberbach polynomials $\{\pi _{n}(z,z_{0})\}$ for the pair $(G,z_{0})$ converge uniformly to $\varphi (z,z_{0})$ on compact subsets of the Jordan domain $G.$ In this paper we study the speed of $\left\| \varphi -\pi _{n}\right\| _{C(B)}\rightarrow 0,$ $n\rightarrow \infty ,$ in domains of the complex plane with a complicated boundary structure.