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Uniform convergence of the p-Bieberbach polynomials in domains with zero angles

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Abstract

Let G ⊂ ℂ be a simply connected domain whose boundary L := ∂G is a Jordan curve and 0 ∈ G. Let w = φ(z) be the conformal mapping of G onto the disk B(0, r 0) := {w : |w| < r 0}, satisfying φ(0) = 0, φ′(0) = 1. We consider the following extremal problem for p > 0:

$\iint_G {|\phi '(z) - P'_n (z)|^p d\sigma _z \to \min }$

in the class of all polynomials P n (z) of degree not exceeding n with P n (0) = 0, P n ′ (0) = 1. The solution to this extremal problem is called the p-Bieberbach polynomial of degree n for the pair (G, 0). We study the uniform convergence of the p-Bieberbach polynomials B n,p (z) to the φ(z) on \(\bar G\) with interior and exterior zero angles determined depending on the properties of boundary arcs and the degree of their “touch”.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Arts and Science, Mersin University, Mersin, 33343, Turkey

    Fahreddin G. Abdullayev & Pelin N. Özkartepe

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  1. Fahreddin G. Abdullayev
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  2. Pelin N. Özkartepe
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Correspondence to Fahreddin G. Abdullayev.

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Abdullayev, F.G., Özkartepe, P.N. Uniform convergence of the p-Bieberbach polynomials in domains with zero angles. Sci. China Math. 58, 1063–1078 (2015). https://doi.org/10.1007/s11425-014-4908-x

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