Differential properties of mappings, conformal at a point
type: Item
instance of: scholarly article
title: Differential properties of mappings, conformal at a point
zbMATH Open document ID: 0597.30022
DOI: 10.1007/BF00969348
author: S. Z. Shefel'
author: Igor G. Nikolaev
published in: Siberian Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: Let f be a quasiconformal mapping of the ball \(B=B(0,r)\) in \(R^ n\) onto a domain D. Suppose that the coefficient K(x) of quasiconformality of f at x satisfies \(K(x)-1\leq c| x|^{m+\alpha}\) a.e. in B, for some \(m=0,1,...\), and \(0<\alpha <1\). Main theorem: There is a function \(g: R^ n\to R^ n\) whose coordinate functions are polynomials with degrees not greater than \(m+1\) such that \[ | f(x)-g(x)| \leq c'| x|^{m+\alpha +1}\quad and\quad c'=c'(c,r,m,\alpha,n,diam(D)). \] For \(n\geq 3\) the proof uses an approximation result, due to Yu. G. Reshetnyak, and, in fact, the result follows from this for \(m=0\). For \(n=2\) special conformal approximations for f at 0 are used.
Mathematics Subject Classification ID: 30C62
zbMATH DE Number: 3962082
zbMATH Keywords: differentiable quasiconformal mappings
reviewed by: O. Martio
MaRDI profile type: Publication
cites work: Q3673610
cites work: On the order of closeness of a spatial quasiconformal mapping to conformal
full work available at URL: https://doi.org/10.1007/bf00969348
OpenAlex ID: W2090078563
description: scientific article; zbMATH DE number 3962082