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# On the Uniform Convergence of the Generalized Bieberbach Polynomials in Regions with K-Quasiconformal Boundary

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## Abstract

Let G be a finite domain in the complex plane with K-quasicon formal boundary, z 0 be an arbitrary fixed point in G and p>0. Let jp ( z ): = òx0 x [ f( z) ]2/8 dz\varphi _p \left( z \right): = \int_{x_0 }^x {\left[ {\phi \left( \zeta \right)} \right]^{2/8} } d\zeta , and let \iintc | jp ( z ) - Px1 (z) |p d0x \iint\limits_c {\left| {\varphi _p \left( z \right) - P_x^1 (z)} \right|^p d0_x } in the class \mathop Õn \mathop \prod \limits_n of all polynomials of degree [`(G)]\bar G in case of $p > 2 - \frac{{K^2 + 1}}{{2K^4 }}$p > 2 - \frac{{K^2 + 1}}{{2K^4 }} .
... (b) Theorems 1.1-1.3 and Corollaries 1.4-1.6, extend the results of [2,4,6,11,12,16,18,19,23,25,30,31] to regions bounded by piecewise quasiconformal curves with interior and exterior zero angles. Corollary 1.6 with β = 0 is the same as in [25]. ...
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We study the uniform and mean convergence of the generalized Bieberbach polynomials in regions having a finite number of interior and exterior zero zero angles.
... In the case p = 2, the existence of a sequence {ε n,p } → 0, n → ∞, that satisfies (1.4) for some domains with quasiconformal piecewise-smooth (without cusps) boundary was investigated in [17,3,6], etc. It is well known that quasiconformal curves have many properties, but they do not have zero angles. ...
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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 Pn ≤ n 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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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| } \lessthan \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.