Sharp Markov brothers type inequality in the spaces \(L_{p}\) and \(L_{1}\) on a closed interval
type: Item
instance of: scholarly article
title: Sharp Markov brothers type inequality in the spaces \(L_{p}\) and \(L_{1}\) on a closed interval
zbMATH Open document ID: 1298.41021
DOI: 10.1134/S0081543812050161
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2014-09-18T00:00:00Z
review text: Let \(\mathcal{P}_n\) be the set of algebraic polynomials of degree exactly \(n\geq 2\) with real coefficients. An equality \(\|P^{(n-1)}\|_p\leq c_p(n)\|P\|_1\) with exact constant \(c_p(n),\;p\in [0,\infty],\) between the \(L_p\)-mean of the \((n-1)\)th derivative of an polynomial \(P\in \mathcal{P}_n\) and \(L_1\)-mean of this polynomial on the interval \([-1,1]\) is studied. Sharp constants and extremal polynomials are established. It is proved that the unique extremal polynomial, up to a multiplicative constant, for \(p\in [0,2n^2-1]\) is the Chebyshev polynomial of the second kind and for \(p\in (2n^2-1,\infty ]\) this is the Zolotareff polynomial in the metric \(L_1[-1,1]\).
reviewed by: D. K. Ugulava
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A44
zbMATH DE Number: 6345607
zbMATH Keywords: Markov type inequality
zbMATH Keywords: \(L_p\)-means
zbMATH Keywords: sharp constants
zbMATH Keywords: extremal polynomials
zbMATH Keywords: Chebyshev polynomial
zbMATH Keywords: Zolotareff polynomial
zbMATH Keywords: Nikol'skii type inequality
author: Ivan E. Simonov
MaRDI profile type: Publication
cites work: The Markov brothers inequality in \(L_0\)-space on an interval
cites work: The sharp Markov-Nikol'skii inequality for algebraic polynomials in the spaces \(L q\) and \(L _{0}\) on a closed interval
cites work: An extension of the Markov inequality
cites work: Q4459954
cites work: Q4858543
cites work: New Inequalities of Markov Type
cites work: On some generalizations of a theorem of A. Markoff
cites work: On the exact constant in the \(L_2\) Markov inequality
cites work: Q4861743
cites work: Q4779788
full work available at URL: s0081543812050161
OpenAlex ID: W2093429064
description: scientific article; zbMATH DE number 6345607