Uniform Convergence of the Generalized Bieberbach Polynomials in Regions with Zero Angles期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Uniform Convergence of the Generalized Bieberbach Polynomials in Regions with Zero Angles

Authors:	F G Abdullayev

Institution:	(1) Faculty of Arts and Science, Department of Mathematics, Mersin University, 33342 Mersin, Turkey

Abstract:	Let C be the extended complex plane; G C a finite Jordan with 0 G; w= (z) the conformal mapping of G onto the disk $B(0;\varrho _0 ): = \{ w:\|w\| < \varrho _0 \}$ normalized by $\varphi (0) = 0 {\text{and}} \varphi '(0) = 1$ . Let us set $\varphi _p (z): = \int_0^z {\left {\varphi '(\zeta )} \right]} ^{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 p}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$p$}}} {\text{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 $\iint\limits_G {\left\| {\varphi '_p (z) - P'_n (z)} \right\|^p {\text{d}}\sigma _{\text{z}} }$ in the class of all polynomials of degree not exceeding n with $P_n (0) = 0,{\text{ }}P'_n (0) = 1$ . In this paper we study the uniform convergence of the generalized Bieberbach polynomials $\pi _{n,p} (z){\text{ to }}\varphi _p (z){\text{ on }}\overline G$ 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
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