Cases of equality for a class of bound-preserving operators over \(P_n\).
type: Item
instance of: scholarly article
title: Cases of equality for a class of bound-preserving operators over \(P_n\).
zbMATH Open document ID: 1051.30010
DOI: 10.1007/BF03321064
published in: Computational Methods and Function Theory
publication date: 2004-09-28T00:00:00Z
review text: A new proof of the well-known inequality \[ | zp'(z) - np(z)| + | zp'(z)| \leq n | p| _D, \quad | z| \leq 1, \tag \(*\) \] is given, where \(p\) is an algebraic polynomial of degree at most \(n\) with complex coefficients and \(| p| _D = \sup_{| z| \leq 1}{| p(z)| }\). All possible cases of the equality in (\(*\)) are described. Also a new proof and all possible cases of equality are given for the inequality \[ | q*p(z)| + | \widetilde q*p(z)| \leq | p| _D, \quad | z| \leq 1, \] where \(q\) is a polynomial of degree at most \(n-1\) with \(q(0)=1\) and \(\inf_{| u| \leq 1} \operatorname{Re} q(u) > \frac{1}{2}\), \(\widetilde q(z) = z^n \overline{q(1/\bar z)}\) and \(*\) denotes the Hadamard product of two polynomials.
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 2104123
zbMATH Keywords: complex polynomials
zbMATH Keywords: Bernstein inequality
zbMATH Keywords: Ruscheweyh inequality
author: Richard Fournier
reviewed by: Elena E. Berdysheva
MaRDI profile type: Publication
cites work: Q3319163
cites work: Q5310474
cites work: Q4779788
full work available at URL: https://doi.org/10.1007/bf03321064
OpenAlex ID: W1985835332
description: scientific article; zbMATH DE number 2104123