Quasiextremal distance domains and integrability of derivatives of conformal mappings
type: Item
instance of: scholarly article
title: Quasiextremal distance domains and integrability of derivatives of conformal mappings
zbMATH Open document ID: 0809.30005
author: John B. Garnett
author: Shanshuang Yang
published in: Michigan Mathematical Journal
publication date: 1995-04-05T00:00:00Z
review text: Let \(f:\Omega \to \mathbb{D}\) be a conformal mapping from a simply connected domain \(\Omega\) into the open unit disk \(\mathbb{D}\). Let \(L\) be any straight line in the plane. The paper deals with the following problem: for which values of \(p\) is \[ \int_{L \cap \Omega} \bigl | f'(z) \bigr |^ pds < \infty? \tag{1} \] By an example due to Baernstein \(f'\) is not \(p\)-integrable for all \(1 \leq p<2\). The paper contains several geometric conditions, some on \(f(L)\) and some on \(\Omega\), which guarantee that (1) holds for all \(1 \leq p<2\). In particular it is proved: if \(\partial \Omega\) is a smooth (i.e. \(C^ 1\) class) Jordan curve or if \(\mathbb{D} \subset \Omega\) and \(f(\partial \mathbb{D})\) has both left and right tangents at each point and is of bounded rotation, then (1) holds for all \(1 \leq p<2\).
reviewed by: Jan Janas
Mathematics Subject Classification ID: 30C20
Mathematics Subject Classification ID: 30C35
zbMATH DE Number: 682551
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029005004
OpenAlex ID: W2069889474
DOI: 10.1307/MMJ/1029005004
description: scientific article; zbMATH DE number 682551