Geodesic
Uniform convergence of the generalized Bieberbach polynomials in regions with zero angles
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 3, pp. 643-660 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 30C10, 30C30, 30C70, 30E10
Keywords: conformal mapping; Quasiconformal curve; Bieberbach polynomials; complex approximation 
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     title = {Uniform convergence of the generalized {Bieberbach} polynomials in regions with zero angles},
     journal = {Czechoslovak Mathematical Journal},
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     url = {http://geodesic.mathdoc.fr/item/CMJ_2001_51_3_a14/}
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Abdullayev, F. G. Uniform convergence of the generalized Bieberbach polynomials in regions with zero angles. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 3, pp. 643-660. http://geodesic.mathdoc.fr/item/CMJ_2001_51_3_a14/

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