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- W2063827189 abstract "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 $$ {varphi_p}(z):=intlimits_0^z {{{{left[ {varphi^{prime}left( zeta right)} right]}}^{2/p }}dzeta } $$ and let π n,p (z) be the generalized Bieberbach polynomial of degree n for the pair (G, 0) that minimizes the integral $$ iintlimits_G {{{{left| {{{{varphi^{prime}}}_p}(z)-{{{P^{prime}}}_n}(z)} right|}}^p}d{sigma_z}} $$ 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." @default.
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- W2063827189 date "2012-10-01" @default.
- W2063827189 modified "2023-09-26" @default.
- W2063827189 title "On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles" @default.
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- W2063827189 doi "https://doi.org/10.1007/s11253-012-0669-2" @default.
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