Markov-type inequalities for polynomials with restricted zeros
type: Item
instance of: scholarly article
title: Markov-type inequalities for polynomials with restricted zeros
zbMATH Open document ID: 0943.41005
author: Yanyan Li
published in: Journal of Approximation Theory
publication date: 2000-08-30T00:00:00Z
review text: Let \(D^{\alpha}\) be a lens shaped, symmetric with respect to the real axis region in \(\mathbb C\), bounded by two circular arcs joining 1 and -1 and meeting each other at an inner angle \(\alpha\pi\) in \(\pm 1\). The author shows the following version of Markov's inequality for polynomials with restrictions on the distribution of the zeros: For every \(0\leq\alpha<1\) there exists a constant \(c(\alpha)\) depending only on \(\alpha\) such that \[ \|p^\prime\|\leq c(\alpha)n^{2-\alpha}\|p\| \] for polynomials \(p\) of degree \(n\) nonvanishing on \(D^\alpha\). Here \(\|\cdot\|=\sup|\cdot|([-1,1])\). Applying this result to the polynomial \(q(z)=\sup_{z\in D^\alpha}|p(z)|+p(z)\) yields, in particular, the well-known Szegö estimate \[ |p^\prime(\pm 1)|\leq c(\alpha)n^{2-\alpha}\sup_{z\in D^\alpha}|p(z)|. \]
reviewed by: Wiesław Pleśniak
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 30A10
Mathematics Subject Classification ID: 26C10
zbMATH DE Number: 1389114
zbMATH Keywords: Markov inequality
zbMATH Keywords: Szegő inequality
zbMATH Keywords: polynomials with restricted zeros
MaRDI profile type: Publication
full work available at URL: jath.1999.3363
OpenAlex ID: W2062754411
cites work: On extremal properties of the derivatives of polynomials
cites work: Q4858543
cites work: Markov-Bernstein-type inequalities for classes of polynomials with restricted zeros
cites work: Markov-Bernstein type inequalities for constrained polynomials with real versus complex coefficients
cites work: Q4320142
cites work: Q3852455
DOI: 10.1006/JATH.1999.3363
description: scientific article; zbMATH DE number 1389114