Conformal mapping of parabola-shaped domains
type: Item
instance of: scholarly article
title: Conformal mapping of parabola-shaped domains
zbMATH Open document ID: 1058.30010
DOI: 10.1007/BF03321059
author: Tom Carroll
author: Walter K. Hayman
published in: Computational Methods and Function Theory
publication date: 2004-09-28T00:00:00Z
review text: Let \(P=\{z=x+iy:x>0,\;| y| <Ax^\alpha\}\), \(0<\alpha<1\), be a parabola-shaped domain and \(S=\{w+u+iv:| v| <\pi/2\}\) be a strip in the complex plane. Suppose that \(f\) is the conformal mapping which maps \(P\) onto \(S\) and is real and increasing for real \(z\). The authors obtain sharp estimates for \(f\) and \(f^\prime\). Precisely, they prove the asymptotic formulae \[ f(z)=\frac{\pi}{2A(1-\alpha)} z^{1-\alpha}+o(1), \] and \[ f^\prime(z)=\left [\frac{\pi}{2A}+o(1)\right ]x^{-\alpha} \] uniformly as \(z\to \infty\) in \(P\). Classical results for more general conformal maps due to Lindelöf and Warschawski are used in the proof. However, in addition, the authors use long arguments associated with the specific map \(f\) to obtain the above asymptotic formulae. These formulae are used in the paper ``Sharp integrability for Brownian motion in parabola-shaped domains'' [to appear in J. Funct. Anal.] by \textit{R. Bañuelos} and \textit{T. Carroll} for the study of the exponential decay of harmonic measure in parabola-shaped domains in the \(n\)-dimensional Euclidean space.
reviewed by: Dimitrios Betsakos
Mathematics Subject Classification ID: 30C20
zbMATH DE Number: 2104118
zbMATH Keywords: conformal mapping
zbMATH Keywords: distortion
zbMATH Keywords: Dini-smooth domain
MaRDI profile type: Publication
cites work: Sharp integrability for Brownian motion in parabola-shaped regions
full work available at URL: https://doi.org/10.1007/bf03321059
OpenAlex ID: W2138905011
description: scientific article; zbMATH DE number 2104118