The maximal range problem for a bounded domain
type: Item
instance of: scholarly article
title: The maximal range problem for a bounded domain
zbMATH Open document ID: 1169.30001
author: Stephan Ruscheweyh
author: V. V. Andrievskii
published in: Journal of Approximation Theory
publication date: 2009-06-30T00:00:00Z
review text: Let \(\Omega\subset\mathbb C\) be a bounded domain, \(0\in\Omega\), \(L:=\partial\Omega\), and let \(\mathcal P_n\) be the set of all complex polynomials of degree at most \(n\). Denote \(\mathcal P_n(\Omega):=\{p\in\mathcal P_n: p(0)=0, p(\mathbb D)\subset\Omega\}\), where \(\mathbb D\) is the unit disk, and \(\Omega_n:=\bigcup_{p\in\mathcal P_n(\Omega)}p(\mathbb D)\) is the maximal range of \(\mathcal P_n(\Omega)\). For a conformal mapping \(f\) of \(\mathbb D\) onto \(\Omega\) with \(f(0)=0\), denote \(f_s(z):=f((1-s)z)\), \(0\leq s<1\). Assume \(f\) is not a polynomial. For a simply connected domain \(\Omega\) denote by \(s(n,\Omega)\) the infimum of \(s\) such that there exists a univalent polynomial \(p\in\mathcal P_n(\Omega)\) with the property \(f_s\prec p\prec f\). An earlier result of the authors states that \[ s(n,\Omega)\leq\frac{c_0}{n}\tag{1} \] for all sufficiently large \(n\) and a universal constant \(c_0>1\). The authors prove Theorem 1 which shows in particular that (1) is sharp for a wide class of bounded domains. Theorem 2 of the article implies that (1) is of the right order if \(L\) has in at least one point something like an acute angle with respect to \(\Omega\). Let \(\Omega(t):=\{z\in\Omega: \text{dist}(z,L)>t\}\), \(t>0\), and \(\Omega^0(t)\) denote the simply connected component of \(\Omega(t)\ni0\) containing the origin. The discrepancy between the domain and its maximal ranges is described by Theorem 3 according to which, for simply connected \(\Omega\), there exists \(n_0=n_0(\Omega)\) and \(c_4=c_4(\Omega)>0\) such that \(\Omega_n\supset\Omega^0(c_4(\log n)^{-1/2})\) for \(n>n_0\). The better estimates for the closeness of \(\Omega_n\) to \(\Omega\) are given in Theorem 4 and Theorem 5 in terms of the Hausdorff distance. Theorem 5 relates to John domains satisfying the \(k\)-quasidisk condition for \(0\leq k<1\). Theorem 6 characterizes the closeness of a boundary point \(z_0\in L\) to \(p(\mathbb D)\), \(p\in\mathcal P_n(\Omega)\) u…
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 5572508
zbMATH Keywords: maximal range
zbMATH Keywords: polynomial
zbMATH Keywords: quasiconformal map
zbMATH Keywords: quasidisk
zbMATH Keywords: John domain
reviewed by: Dmitri V. Prokhorov
MaRDI profile type: Publication
full work available at URL: j.jat.2008.08.002
OpenAlex ID: W1985029298
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DOI: 10.1016/J.JAT.2008.08.002
description: scientific article; zbMATH DE number 5572508