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On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles

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Ukrainian Mathematical Journal Aims and scope

Let ℂ be the complex plane, let \( \bar{\mathbb{C}}=\mathbb{C}\cup \left\{ \infty \right\} \), let G ⊂ ℂ be a finite Jordan domain with 0 ∈ G; let L := ∂G; let Ω := \( \bar{\mathbb{C}}\backslash \bar{G} \), and let w = φ(z) be a conformal mapping of G onto a disk B(0, ρ 0) := \( \left\{ {w:\left| w \right|<{\rho_0}} \right\} \) normalized by the conditions φ(z) = 0 and \( {\varphi}^{\prime}(0)=1 \), where ρ 0 = ρ 0(0, G) is the conformal radius of G with respect to 0. Let

$$ {\varphi_p}(z):=\int\limits_0^z {{{{\left[ {\varphi^{\prime}\left( \zeta \right)} \right]}}^{2/p }}d\zeta } $$

and let π n,p (z) be the generalized Bieberbach polynomial of degree n for the pair (G, 0) that minimizes the integral

$$ \iint\limits_G {{{{\left| {{{{\varphi^{\prime}}}_p}(z)-{{{P^{\prime}}}_n}(z)} \right|}}^p}d{\sigma_z}} $$

in the class of all polynomials of degree deg Pnn such that Pn(0) = 0 and \( {{P^{\prime}}_n}(0)=1 \). We study the uniform convergence of the generalized Bieberbach polynomials π n,p (z) to φ p (z) on \( \bar{G} \) with interior and exterior zero angles determined depending on properties of boundary arcs and the degree of their tangency. In particular, for Bieberbach polynomials, we obtain improved estimates for the rate of convergence in these domains.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 5, pp. 582–596, May, 2012.

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Abdullayev, F.G., zkartepe, N.P.Ö. On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles. Ukr Math J 64, 653–671 (2012). https://doi.org/10.1007/s11253-012-0669-2

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