On Schwarz-Christoffel mappings
type: Item
instance of: scholarly article
title: On Schwarz-Christoffel mappings
zbMATH Open document ID: 1312.30001
author: Christian Pommerenke
author: Martin Chuaqui
published in: Pacific Journal of Mathematics
publication date: 2014-10-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1207.6143
reviewed by: Janusz Godula
Mathematics Subject Classification ID: 30C20
Mathematics Subject Classification ID: 30C35
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30J10
zbMATH DE Number: 6361405
zbMATH Keywords: Schwarz-Christoffel mapping
zbMATH Keywords: Blaschke product
MaRDI profile type: Publication
OpenAlex ID: W2063651956
arXiv ID: 1207.6143
DOI: 10.2140/PJM.2014.270.319
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review text: Let \(f\) be a Schwarz-Christoffel mapping of the unit disk \(\mathbb D\) onto the interior of an \((n+1)\)-gon. The first author et al. [Ann. Acad. Sci. Fenn., Math. 36, No. 2, 449--460 (2011; Zbl 1239.30003); Proc. Am. Math. Soc. 140, No. 10, 3495--3505 (2012; Zbl 1283.30048)] have shown that the pre-Schwarzian of \(f\) has the form NEWLINE\[NEWLINE\frac{f''}{f'}=\frac{2\frac{B_1}{B_2}}{1-z\frac{B_1}{B_2}}NEWLINE\]NEWLINE for some finite Blaschke products \(B_1\), \(B_2\) without common zeros, with respective degrees \(d_1,\;d_2\) satisfying \(d_1+d_2=n\). It is known that NEWLINE\[NEWLINE\frac{f''}{f'}=-2\sum_{k=1}^{n+1}\frac{\beta_k}{z-z_k},\tag{1}NEWLINE\]NEWLINE where \(z_k\) are prevertices and \(2\pi\beta_k\) are the exterior angles ot \(z_k\). Theorem 2 says that \(d_2\) is equal to the number of concave vertices, while \(d_1+1\) is equal to the number of convex vertices. In Theorem 3 the authors characterize the mappings \(f\) onto the exterior of an \((n+2)\)-gon. Theorem 4 solves the problem of the univalence of solutions of (1). Namely, let \(0\leq t_1<\cdots <t_{n+1}<2\pi\), \(z_k=e^{it_k}\), \(\beta_k\in\mathbb R\) and \(\sum_{k=1}^{n+1} \beta_k=1\), \(\sum_{k=1}^{n+1} |\beta_k|\leq 2\). Then \(f\) defined by NEWLINENEWLINENEWLINE\[NEWLINEf(z)=a\prod_{k=1}^{n+1}(z-z_k)^{-2\beta_k},\quad a\in\mathbb C\setminus\{0\},NEWLINE\]NEWLINE is univalent in \(\mathbb D\).NEWLINENEWLINEIn the final part of the paper the authors study the geometric interplay between the location of the zeros of the Blaschke products and the separation of the prevertices.
description: scientific article; zbMATH DE number 6361405