Approximation properties of the Bieberbach polynomials in closed Dini-smooth domains

Abstract

Let $G$ be a finite Dini-smooth domain and $% w=\varphi _{0}(z)$ be the conformal mapping of $G$ onto $D\left( 0,r_{0}\right) :=\{w:\mid w\mid <r_{0}\}$ with the normalization $ \varphi _{0}(z_{0})=0$, $\varphi _{0}^{^{\prime }}(z_{0})=1$, where $z_{0}\in G$. We investigate the approximation properties of the Bieberbach polynomials $ \pi _{n}(z)$, $ n=1,2,3, \cdots $ for the pair $\left( G,z_{0}\right) $ and estimate the error \begin{equation*} \parallel \varphi _{0}-\pi _{n}\parallel _{\overline{G}} := \max \{\mid \varphi _{0}\left( z\right) -\pi _{n}\left( z\right) \mid :z\in \overline{G} \} \end{equation*} in accordance with the geometric parameters of $\overline{G}$.