Mersin University, Faculty of Arts and Science, Department of Mathematics, 33342 Mersin, Turkey e-mail: f.g.abdullayev@usa.net, fabdul@mersin.edu.tr
Abstract: 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\{ {w {\left| w\right| }<\rho_0} \right\} }$ normalized by $\varphi(0)=0$ and ${\varphi}'(0)=1$. Let us set $\varphi_p(z):=\int_0^z{{\left[ {{\varphi} '(\zeta)}\right] }^{2/p}}\dd\zeta$, and let $\oldpi_{n,p}(z)$ 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'(z)}\right| }}^p\dd\sigma_z$ in the class of all polynomials of degree not exceeding $\leq n$ with $P_n(0)=0$, $P_n'(0)=1$. In this paper we study the uniform convergence of the generalized Bieberbach polynomials $\oldpi_{n,p}(z)$ 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.
Keywords: conformal mapping, Quasiconformal curve, Bieberbach polynomials, complex approximation
Classification (MSC 2000): 30C30, 30E10, 30C70
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