On the length of lemniscates
type: Item
instance of: scholarly article
title: On the length of lemniscates
zbMATH Open document ID: 0958.30002
author: Alexandre Eremenko
author: Walter K. Hayman
published in: Michigan Mathematical Journal
publication date: 2001-01-01T00:00:00Z
review text: For a monic polynomial \(p\) of degree \(d\), we write \(E(p)=\{z:|p(z)|=1\}\). \textit{P. Erdős, F. Herzog}, and \textit{C. Piranian} [J. Anal. Math. 6, 125-148 (1958; Zbl 0088.25302)] conjectured that the length \(|E(p)|\) is maximal when \(p(z)=z^d+1\). Recently \textit{P. Borwein} [Proc. Am. Math. Soc. 123, No. 3, 797-799 (1995; Zbl 0821.31002)] proved that \(|E(p)|\leq 68.32 d.\) In the paper under review it is shown that \(|E(p)|\leq 9.173 d.\) The authors also prove that extremal polynomials exist, and that there is an extremal polynomial \(p_o\) such that all critical points of \(p_o\) lie in \(E(p_o)\) and \(E(p_o)\) is connected. A similar problem for rational functions is also studied and completely solved. In the proofs the authors make ingenious use of a variety of tools: quasiconformal maps, subharmonic functions, logarithmic capacity, the Riemann-Hurwitz theorem, integral-geometric formulae, and special properties of complex polynomials.
reviewed by: Dimitrios Betsakos
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 30A10
Mathematics Subject Classification ID: 30C85
zbMATH DE Number: 1548313
zbMATH Keywords: polynomial
zbMATH Keywords: rational function
zbMATH Keywords: logarithmic capacity
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1030132418
OpenAlex ID: W2160256761
DOI: 10.1307/MMJ/1030132418
Recommended article: Q4410045
Recommended article: On estimates of lengths of lemniscates
Recommended article: Length functions of lemniscates
Recommended article: On a geometric property of Lemniscates
Recommended article: Q3682730
Recommended article: Lengths of lemniscates. Variations of rational functions
Recommended article: On the Convexity of Lemniscates
Recommended article: Area and the inradius of lemniscates
Recommended article: On lemniscate functions
description: scientific article; zbMATH DE number 1548313