On the exact Markov inequality for \(k\)-monotone polynomials in uniform and \(L_1\)-norms
type: Item
instance of: scholarly article
title: On the exact Markov inequality for \(k\)-monotone polynomials in uniform and \(L_1\)-norms
zbMATH Open document ID: 1224.41044
published in: Acta Mathematica Hungarica
publication date: 2011-02-08T00:00:00Z
review text: The authors consider the classical extremal problem of estimating norms of higher order derivatives of algebraic polynomials when their norms are given. The corresponding extremal problem for general polynomials in uniform norm was solved by A. A. Markov, while Bernstein found the exact constant in the Markov inequality for monotone polynomials. In the paper, the authors give Markov-type inequalities for higher order derivatives in the general class of \(k\)-monotone polynomials. Specifically, the authors consider the extremal problem of finding the maximal value of the uniform and \(L_r\)-norm of the \(i\)-th derivatives of normalized \(k\)-monotone polynomials \[ M^{(i)}(n,k)_r:=\sup _{p\in {P}_n^{(k)}}{\| p^{(i)}\| _r \over \| p\| _r},\quad 0\leq k\leq n,\;1\leq i\leq k,\;0<r\leq \infty, \] where \(P_n^{(k)}\) denotes the space of algebraic polynomials \(p\) of degree at most \(n\) such that \(p^{(j)}(x)\leq 0\), \(| x| \leq 1\), \(0\leq j\leq k\) and \(\| \cdot \| _r\) stands for the usual \(L_r\)-norm on \([-1,1]\). The exact value \(M^{(1)}(n,0)_{\infty }=n^2/2\) was found by A. A. Markov. Bernstein proved that \(M^{(1)}(n,1)_{\infty }\) equals \((n+1)^2/8\) or \(n(n+2)/8\) if \(n\) is odd or even, respectively. The authors consider the above problem for the general set of \(k\)-monotone polynomials \({P}_n^{(k)}\) for any \(2\leq k\leq n\), \(1\leq i<k\) and \(r=1\) or \(\infty\). For the case \(i=1\), they find the exact value of the Markov constant form \(k\)-monotone polynomials which will be given in terms of the largest zero of certain Jacobi polynomials. They also give the exact asymptotics for \(M^{(1)}(n,k)_{p}\) (\(r=1\) or \(\infty \)) with respect to \(n\) and \(k\). For \(i>1\), they give upper and lower estimates for \(M^{(i)}(n,k)_{\infty }\) in terms of zeros of Jacobi polynomials. In particular, the results imply the somewhat surprising fact that the order of Markov factors for \(k\)-monotone polynomials is \(n^2/k\).
reviewed by: Daniel Cárdenas Morales
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 5848550
zbMATH Keywords: Markov inequality
zbMATH Keywords: \(k\)-convex polynomial
zbMATH Keywords: Jacobi polynomial
zbMATH Keywords: Bessel function
author: András Kroó
author: Jószef Szabados
MaRDI profile type: Publication
full work available at URL: s10474-009-8219-6
OpenAlex ID: W2009212467
cites work: Q4122196
cites work: Q4779788
DOI: 10.1007/S10474-009-8219-6
description: scientific article; zbMATH DE number 5848550