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Keywords:
conformal mapping; Quasiconformal curve; Bieberbach polynomials; complex approximation
conformal mapping; Quasiconformal curve; Bieberbach polynomials; complex approximation
Summary:
Let $C$ be the extended complex plane; $G\subset C$ a finite Jordan with $ 0\in G$; $w=\varphi (z)$ the conformal mapping of $G$ onto the disk $ B\left( {0;\rho _{0}}\right):={\left\rbrace {w\:{\left| {w}\right| }<\rho _{0}} \right\lbrace }$ normalized by $\varphi (0)=0$ and ${\varphi }^{\prime }(0)=1$. Let us set $\varphi _{p}(z):=\int _{0}^{z}{{\left[ {{\varphi } ^{\prime }(\zeta )}\right] }^{{2}/{p}}}\mathrm{d}\zeta $, and let $\pi _{n,p}(z)$ be the generalized Bieberbach polynomial of degree $n$ for the pair $(G,0)$, which minimizes the integral $ \iint \limits _{G}{{\left| {{\varphi }_{p}^{\prime }(z)-{P}_{n}^{\prime }(z)}\right| }}^{p}\mathrm{d}\sigma _{z}$ in the class of all polynomials of degree not exceeding $\le n$ with $P_{n}(0)=0$, ${P}_{n}^{\prime }(0)=1$. In this paper we study the uniform convergence of the generalized Bieberbach polynomials $\pi _{n,p}(z)$ to $\varphi _{p}(z)$ on $\overline{G}$ with interior and exterior zero angles and determine its dependence on the properties of boundary arcs and the degree of their tangency.
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