A Markov-type inequality for the derivatives of constrained polynomials
type: Item
instance of: scholarly article
title: A Markov-type inequality for the derivatives of constrained polynomials
zbMATH Open document ID: 0722.41013
DOI: 10.1016/0021-9045(90)90124-9
author: Tamás Erdélyi
published in: Journal of Approximation Theory
publication date: 1990-01-01T00:00:00Z
review text: The magnitude of \[ \sup_{p\in S^ m_ n(r)}(\max_{-1\leq x\leq 1}| p'(x)| /\max_{-1\leq x\leq 1}| p(x)|) \] is examined, where \(S^ m_ n(r)\) (0\(\leq m\leq n\), \(0<r\leq 1)\) denotes the set of those real algebraic polynomials of degree at most n which have all but at most m zeroes outside the circle \(K(r)=\{z\in {\mathbb{C}}:| z| <r\}\). It is proved that \[ c_ 1(m)(n+(1-r)n^ 2)\leq \sup_{p\in S^ m_ n(r)}(\max_{-1\leq x\leq 1}| p'(x)| /\max_{-1\leq x\leq 1}| p(x)|)\leq c_ 2(m)(n+(1-r)n^ 2) \] holds for every \(0<r\leq 1\) and \(0\leq m\leq n\), where \(c_ 1(m)\) and \(c_ 2(m)\) depend only on m. This estimate is a generalization of Markov's inequality \((r=0\), \(m=0)\) and a result due to Lorentz \((r=1\), \(m=0)\).
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 4190350
zbMATH Keywords: Markov's inequality
reviewed by: Margareta Heilmann
MaRDI profile type: Publication
full work available at URL: 0021-9045(90)90124-9
OpenAlex ID: W2125102479
cites work: Markov's Inequality for Polynomials with Real Zeros
cites work: Q3761020
cites work: Markov-type estimates for certain classes of constrained polynomials
cites work: Markov type estimates for derivatives of polynomials of special type
cites work: The degree of approximation by polynomials with positive coefficients
cites work: Inequalities for derivatives of polynomials of special type
cites work: Q3948105
description: scientific article; zbMATH DE number 4190350