Majorization of the critical points of a polynomial by its zeros
type: Item
instance of: scholarly article
title: Majorization of the critical points of a polynomial by its zeros
zbMATH Open document ID: 1057.30006
DOI: 10.1007/BF03321027
published in: Computational Methods and Function Theory
publication date: 2004-11-01T00:00:00Z
review text: Let \(z_{1},\ldots,z_n\) be the zeros of a polynomial \(f(z)\) and let \(\zeta_{1},\ldots,\zeta_n\) be those of \(zf'(z)\). Suppose that for both polynomials the zeros are labelled in order of non-increasing modulus. The author shows that \[ \sum\limits^k_{\nu=1}| \zeta_\nu| \leq\sum\limits^k_{\nu=1}| z_\nu| ,\qquad k=1,\ldots,n, \] which means that the moduli of the zeros of \(f(z)\) weakly majorize those of \(zf'(z)\). This refines the Gauss-Lucas Theorem, which states that the convex hull of the zeros of a polynomial \(f\) contains all the critical points of \(f\). Moreover, this weak majorization is preserved if one replaces \(| \zeta_\nu| \) by \(\psi(| \zeta_\nu| )\) and \(| z_\nu| \) by \(\psi(| z_\nu| )\) for \(\nu=1,\ldots,n,\) where \(\psi \circ \exp \) is any non-decreasing convex function on \(\mathbb{R}\). Actually, more general results are established which hold for a polynomial \(f\) and a certain multiplicative composition which may be interpreted as a Hadamard product of \(f\) with a polynomial from a certain class.
reviewed by: W. Koepf
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 2112928
zbMATH Keywords: majorization
zbMATH Keywords: zeros of polynomials
zbMATH Keywords: critical points
zbMATH Keywords: Gauss-Lucas Theorem
zbMATH Keywords: inequalities
author: Gerhard Schmeisser
MaRDI profile type: Publication
cites work: On the zeros of a polynomial and its derivative
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cites work: Inequalities between the Two Kinds of Eigenvalues of a Linear Transformation
full work available at URL: https://doi.org/10.1007/bf03321027
OpenAlex ID: W2064724297
description: scientific article; zbMATH DE number 2112928