Uniformization by rectangular domains: A path from slits to squares

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:486

Article

Solynin, Alexander Yu.¹  Vidanage, Nadeesha C.¹ 

[1] Texas Tech Univ, Dept Math & Stat, Box 41042, Lubbock, TX 79409 USA

关键词: Conformal mapping;    Canonical domain;    Extremal problem;    Rectangle;    Uniformization;   

DOI  :  10.1016/j.jmaa.2020.123927

来源: Elsevier

【 摘 要 】

Let Sigma(Omega) be the class of functions f(z) = z + a(1)/z + ... univalent on a finitely connected domain Omega, infinity is an element of Omega subset of (C) over bar. By a classical result due to H. Grotzsch, the function f(0) maximizing R-a1 over the class Sigma(Omega) maps Omega onto (C) over bar slit along horizontal segments. Recently, M. Bonk found a similar extremal problem, which maximizer f(1) is an element of Sigma(Omega) maps Omega onto a domain on (C) over bar, whose complementary components are squares. In this note, we discuss a parametric family of extremal problems on the class Sigma(Omega) with maximizers f(m), 0 < m < 1, mapping Omega onto domains on (C) over bar, whose complementary components are rectangles with horizontal and vertical sides and with module m. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

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