Area and the inradius of lemniscates
type: Item
instance of: scholarly article
title: Area and the inradius of lemniscates
zbMATH Open document ID: 1166.30020
author: A. Yu. Solynin
author: Alexander S. Williams
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-05-19T00:00:00Z
review text: Let \(p\) be a complex polynomial of degree \(n\), \(c\) a positive constant, \(E(p,c)\) the filled lemniscate of \(p\) of radius \(c\), i.e. \(E(p,c)=\{z\in\mathbb{C}:| p(z)|\leq c\}\) and \(L(E)\) the perimeter of \(E\). An open problem posed by Erdös et al. in 1958 is to maximize the length \(L(E(p,1))\) over all monic polynomials \(p_n\). Erdös conjectured that \(p_n(z)=z^n-1\) provides the desired maximum. In the present paper the authors study the similar problem on the maximum of the ratio of the area of \(E\) divided by \(\pi r^2\), where \(r\) denotes the inradius of \(E\), and prove that it is bounded by some constant depending only of the degree \(n\) for all \(p_n\) and all \(0<c<\infty\). This proves a conjecture posed by \textit{H. H. Cuenya} and \textit{F. E. Levis} [J. Math. Anal. Appl. 336, No. 2, 953--961 (2007; Zbl 1123.32006)].
reviewed by: Jacek Gilewicz
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 5555208
zbMATH Keywords: polynomial
zbMATH Keywords: lemniscate
zbMATH Keywords: area
zbMATH Keywords: inradius
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.01.012
OpenAlex ID: W2051482742
cites work: Pólya-Type Polynomial Inequalities in LPSpaces and Best Local Approximation
cites work: A property of the planar measure of the lemniscates
cites work: Metric properties of polynomials
cites work: Q5574105
cites work: Q4779788
cites work: Q4787437
DOI: 10.1016/J.JMAA.2009.01.012
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description: scientific article; zbMATH DE number 5555208