A conformal invariant for nonvanishing analytic functions and its applications
type: Item
instance of: scholarly article
title: A conformal invariant for nonvanishing analytic functions and its applications
zbMATH Open document ID: 1121.30004
published in: Michigan Mathematical Journal
publication date: 2007-04-05T00:00:00Z
review text: The authors of this paper study a new conformal invariant defined for nonvanishing analytic functions. More precisely, let \(\varphi:D \rightarrow\mathbb{C}^*=\mathbb{C}\backslash \{0\}\) be an analytic function defined on a hyperbolic domain \(D\) with hyperbolic density \(\rho_D(z)\), and let \[ V_D(\varphi) = \sup_{z\in D}\rho_D(z)^{-1}\left| \frac{\varphi'(z)}{\varphi(z)}\right| \, . \] This quantity is conformally invariant in the sense that \(V_D(\varphi)=V_{D_0}(\varphi\circ f)\) for a conformal (univalent, onto) map \(f:D_0\rightarrow D\). In general, \(V_D\) decreases under composition \(\varphi\circ g\) with analytic maps \(g\). The authors study a number of properties of the invariant \(V_D\) and its connection with other relevant analytic quantities related to the geometry of domains and with the behavior of analytic functions defined on them. In particular, they relate \(V_D\) to the circular width (with respect to the origin) \[ W(\Omega)= \left(\text{ inf}_{w\in \Omega}| w| \rho_{\Omega}(w)\right)^{-1} \] of a hyperbolic domain \(\Omega\) not containing the origin. The circular width is not easy to compute, but the authors establish several important properties of \(W(\Omega)\) as well as a geometric characterization of when it is infinite. They also provide a series of examples when the circular width can be determined explicitly. Finally, the authors study the connection of \(V_D\) with the concept of uniform perfectness, relevant in the study of univalent functions and quasiconformal extensions.
Mathematics Subject Classification ID: 30C35
zbMATH DE Number: 5140640
zbMATH Keywords: conformal invariant
zbMATH Keywords: nonvanishing analytic function
zbMATH Keywords: hyperbolic domain
zbMATH Keywords: hyperbolic metric
zbMATH Keywords: circular width
zbMATH Keywords: uniformly perfect domain
zbMATH Keywords: univalent function
author: Yong Chan Kim
author: Toshiyuki Sugawa
reviewed by: Martin Chuaqui
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1156345602
OpenAlex ID: W2072759369
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DOI: 10.1307/MMJ/1156345602
description: scientific article; zbMATH DE number 5140640