Some maximal polynomials must be nonreal
type: Item
instance of: scholarly article
title: Some maximal polynomials must be nonreal
zbMATH Open document ID: 0887.30008
author: Michael Miller
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-05-25T00:00:00Z
review text: Let \(n>1\) and let \(\beta\) be a complex number of modulus not exceeding 1. \(S(n,\beta)\) is the set of complex polynomials of degree \(n\) whose roots all lie in the closed unit disk and at least one root is \(\beta\). For a polynomial \(P\) define \(|P|_\beta\) to be the distance between \(\beta\) and the closest root to \(\beta\) of the derivative \(P'\). Sendov's conjecture (sometimes called the Sendov-Ilyeff conjecture) states that for \(P\in S(n,\beta)\), \(|P|_\beta\leq 1\). The present author introduced and studied extremal polynomials related to the above conjecture. These are polynomials \(P\in S(n,\beta)\) for which \(|P |_\beta\) is maximal. Without loss of generality one can assume that \(0\leq \beta\leq 1\). In the present paper the author proves that if \(P\in S(6,\beta)\) is extremal \(P(z)\) cannot be real. This assertion is a result of the following two results obtained by somewhat lengthy technical estimates. Theorem 1. For all real polynomials \(P\in S(6,\beta)\), \[ |P|_\beta\leq 1- {11\over 30} (1-\beta)+ {1\over 30} (1-\beta)^2 +O(1-\beta)^3. \] Theorem 3. There are polynomials \(P\in S(6, \beta)\) for which \[ |P|_\beta=1- {11\over 30} (1-\beta)+ {49\over 900} (1-\beta)^2 +O(1-\beta)^3. \]
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 1115514
reviewed by: Zalman Rubinstein
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5629
OpenAlex ID: W1968118147
cites work: THE SENDOV CONJECTURE FOR POLYNOMIALS WITH AT MOST SEVEN DISTINCT ZEROS
cites work: Corrigendum: ‘on The Sendov‐Ilyeff Conjecture’
cites work: A note on Ilyeff-Sendov conjecture for polynomials of arbitrary degree with multiple zeros
cites work: Maximal Polynomials and the Ilieff-Sendov Conjecture
cites work: On a problem of Ilyeff
DOI: 10.1006/JMAA.1997.5629
description: scientific article; zbMATH DE number 1115514