Transformation of the quasiconformal reflection coefficient and of the Fredholm eigenvalue
type: Item
instance of: scholarly article
title: Transformation of the quasiconformal reflection coefficient and of the Fredholm eigenvalue
zbMATH Open document ID: 1297.30040
published in: Georgian Mathematical Journal
publication date: 2014-06-20T00:00:00Z
review text: Let \(\gamma\) be a closed Jordan curve in the complex plane. The reflection coefficient \(q \in [0,1]\) of \(\gamma\) is the smallest value for which a \((1+q)/(1-q)\)-quasiconformal reflection exists and the reciprocal Fredholm eigenvalue \(\kappa \in [0,1]\) could be defined as \(\kappa = \sup(|D_G(u) - D_{G^*}|)/(D_G(u) + D_{G^*})\) where \(G\) and \(G^*\) are the interior and exterior domains of \(\gamma\), respectively, and \(D\) refers to the Direchlet integral of \(u\) over the marked domain. The supremum is taken over all functions continuous in \(\overline{\mathbb C}\) and harmonic in \(G \cup G^*\), see [\textit{S. Krushkal}, in: Analysis and Mathematical Physics, Trends in Mathematics, Birkhäuser. 349--368 (2009; Zbl 1297.30011)]. The fundamental Ahlfors inequality says that \(\kappa \leq q\) but for many curves there is equality. The author studies the behavior of these quantities under the square root transformation. He assumes that \(\gamma\) is the image of the unit circle under the hydrodynamically normalised univalent mapping \(f(z) = z + a_0 + a_1/z + ...\) of \(|z| > 1\) and \(f\) is analytic on \(|z| = 1\) with \(f'(z) \neq 0\). The square root transformation \(\sqrt{f(z^2)}\) produces from \(\gamma\) a Jordan curve \(\gamma^*\) centrally symmetric with respect to \(0\). Let \(q^*\) and \(\kappa^*\) denote the aforementioned quantities associated with \(\gamma^*\). The main results are: \(q^* \geq q\) and if \(q= \kappa\) and \(q^* = q\), then \(\kappa^* = \kappa\). The paper contains several interesting examples and a detailed study of the case \(\kappa^* = \kappa\).
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30C25
Mathematics Subject Classification ID: 31A05
zbMATH DE Number: 6305834
zbMATH Keywords: quasiconformal reflection coefficient
zbMATH Keywords: Fredholm eigenvalue
zbMATH Keywords: quasiconformal mapping, quasicircle
author: Reiner Kühnau
reviewed by: O. Martio
MaRDI profile type: Publication
full work available at URL: gmj-2014-0017
OpenAlex ID: W2330237063
DOI: 10.1515/GMJ-2014-0017
description: scientific article; zbMATH DE number 6305834