On the inequalities of Bernstein-Markoff for an interval
type: Item
instance of: scholarly article
title: On the inequalities of Bernstein-Markoff for an interval
zbMATH Open document ID: 0596.30003
DOI: 10.1007/BF02790176
author: Clément Frappier
published in: Journal d'Analyse Mathématique
publication date: 1984-01-01T00:00:00Z
review text: Interesting theorems on relationships between the absolute value of polynomials in a certain class, and the absolute value of their derivatives, are derived. These theorems are extensions and refinements of results by A. Markoff, W. Markoff, Duffin and Schaeffer, Schur, and others. The following is an example to the kind of theorems which are obtained: Let \(S_ n\) be the class of polynomials \(P(z)=\sum^{n}_{j=0}a_ jz^ j\), of degree at most n. Let \(P\in S\), \(P(c)=0\) where \(1\leq c\leq (2 \cos^ 2(\pi /4n)/\cos^ 2(\pi /2n))-1,\) and \(| P(x)| \leq 1\quad for\quad x=(b+\cos (\nu \Pi /n))/a,\quad 1\leq \nu \leq n,\) where \(a=[2/(c+1)]\cos^ 2 \Pi /4n;\quad b=[(c-1)/(c+1)]\cos^ 2 \Pi /4n+\sin^ 2 \Pi /4n.\) Then: \[ | p^{(k)}(x)| \leq ([2/(c+1)]\cos^ 2 \Pi /4n)^ k\frac{n^ 2(n^ 2-1^ 2)...(n\quad^ 2-(k-1)^ 2)}{1.3....(2k-1)} \] for -1\(\leq x\leq 1\) and \(k=1,...,n\). If, in addition, P(z) has real coefficients, then \[ | P^{(k)}(x+iy)| \leq ([2/(c+1)]\quad \cos^ 2 \Pi /4n)^ k\quad | T_ n^{(k)}(-1+ayi)| \] for -1\(\leq x\leq 1\), \(-\infty <y<\infty\) and \(k=1,...,n\).
Mathematics Subject Classification ID: 30A10
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 30C10
zbMATH DE Number: 3959852
zbMATH Keywords: inequality of Bernstein-Markov
MaRDI profile type: Publication
cites work: A Refinement of an Inequality of the Brothers Markoff
cites work: Q5774836
cites work: Extremal Properties of the Successive Derivatives of Polynomials and Rational Functions.
cites work: Inequalities for Polynomials on the Unit Interval
cites work: Inequalities for derivatives of polynomials of special type
description: scientific article; zbMATH DE number 3959852