Bound-preserving operators and Bernstein type inequalities
type: Item
instance of: scholarly article
title: Bound-preserving operators and Bernstein type inequalities
zbMATH Open document ID: 1061.41007
DOI: 10.1007/BF03321857
published in: Computational Methods and Function Theory
publication date: 2004-09-27T00:00:00Z
review text: Let \(p(z):=\sum_{k=0}^na_kz^k\) be a complex polynomial of degree at most \(n\). According to the celebrated inequality of Bernstein (*) \(| | p^{\prime}| | \leq n| | p| | \), where \(| p| :=\max_{| z| =1}| p(z)| \) and \(n\geq 1\). For various extensions of inequality (*), see, for example, \textit{Q. I. Rahman} and \textit{G. Schmeisser} [Analytic theory of polynomials, (Oxford University Press, Oxford), 2002; Zbl 1072.30006)]. In the paper under review, the authors are concerned with the case when equality holds in a more general version of Bernstein's inequality. This problem has been open since 1982. The authors also prove an inequality of Bernstein type which, under some additional condition, improves the upper bound for \(| | p^{\prime}| | \).
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 30A10
zbMATH DE Number: 2103475
zbMATH Keywords: polynomials
zbMATH Keywords: bound-preserving operators
zbMATH Keywords: Bernstein type inequalities
author: Dimiter P. Dryanov
author: Richard Fournier
reviewed by: George Csordas
MaRDI profile type: Publication
cites work: Q3319163
cites work: Q5310474
cites work: New Inequalities for Polynomials
cites work: Q3247378
cites work: Second order differential inequalities in the complex plane
cites work: Periodic perturbations of an isochronous center.
cites work: On the convolution of analytic functions.
cites work: Q4084504
cites work: Q5835296
full work available at URL: https://doi.org/10.1007/bf03321857
OpenAlex ID: W2085225125
description: scientific article; zbMATH DE number 2103475