An inequality for derivatives of polynomials with positive coefficients
type: Item
instance of: scholarly article
title: An inequality for derivatives of polynomials with positive coefficients
zbMATH Open document ID: 0814.30005
published in: Journal of Approximation Theory
publication date: 1995-06-18T00:00:00Z
review text: A Bernstein type inequality is proved for the class \(P^ +_ N\) of polynomials of degree at most \(N\) whose coefficients are nonnegative. Let \(n(r,f)\) count the number of zeros with multiplicity of the polynomial \(f\) in the disk \(| z|<r\) and let \(M(r,f)= \max_{| z|=r} | f(z)|\). Then, if \(f\in P^ +_ N\) for \(N=1,2,3,4\), the bound \(M(r, f')\leq [(N+ n(r,f))/ 2r] M(r,f)\) holds for \(r>0\). Furthermore, let \(5m\leq N< 5(m+1)\), \(m=1,2,3,\dots\;\). Then there is a polynomial \(f\in P^ +_ N\) for which \(M(3,f')= [(N+ n(3,f)+ 2m/7) /6] M(3,f)> [(N+ n(3,f)) /6] M(3,f)\).
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 720672
zbMATH Keywords: Bernstein inequality
zbMATH Keywords: coefficients
author: S. P. Zhou
reviewed by: Peter A. McCoy
MaRDI profile type: Publication
full work available at URL: jath.1995.1010
OpenAlex ID: W1973047776
DOI: 10.1006/JATH.1995.1010
Recommended article: Q3475780
Recommended article: Q4689082
Recommended article: Some inequalities for derivatives of polynomials
Recommended article: Some Inequalities for the Derivative of Polynomials
Recommended article: Q4556506
Recommended article: Q4852416
Recommended article: Q3797446
Recommended article: Inequalities concerning the derivative of polynomials
Recommended article: Inequalities for the Derivative of a Polynomial
Recommended article: Inequalities for the derivative of a polynomial
description: scientific article; zbMATH DE number 720672