On a theorem of Bernstein
type: Item
instance of: scholarly article
title: On a theorem of Bernstein
zbMATH Open document ID: 0589.30001
DOI: 10.1216/RMJ-1985-15-4-765
author: B. George
published in: Rocky Mountain Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: The case of equality in the following theorem of Bernstein has been proved. Theorem B. Let P(z) and Q(z) be polynomials satisfying the conditions that Q(z) has all its zeros in \(| z| \leq 1\) and the degree of P(z) does not exceed that of Q(z). If \[ (1)\quad | P(z)| \leq | Q(z)| \quad on\quad | z| =1 \] then \[ (2)\quad | P'(z)| \leq | Q'(z)| \quad on\quad | z| =1. \] In this note it is proved that if there is equality in (2) at any point \(\mu\) on \(| z| =1\), where \(Q(\mu)\neq 0\) then \(P(z)=\alpha Q(z)\), \(| \alpha | =1\).
Mathematics Subject Classification ID: 30A10
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 3946519
zbMATH Keywords: Bernstein inequality
MaRDI profile type: Publication
full work available at URL: rmj-1985-15-4-765
OpenAlex ID: W1973195713
description: scientific article; zbMATH DE number 3946519