Equivalence of the global and local Markov inequalities in the complex plane
type: Item
instance of: scholarly article
title: Equivalence of the global and local Markov inequalities in the complex plane
zbMATH Open document ID: 1414.41006
author: Raimondo Eggink
author: Leokadia Białas-Cież
published in: Advances in Mathematics
publication date: 2019-02-13T00:00:00Z
review text: A compact set $E$ in the complex plane is said to admit the \textit{global Markov inequality} $\text{GMI}(k)$, $k\geq1$, if there exists a constant $M\geq1$ such that \[ {\|p'\|}_{E}\leq Mn^k{\|p\|}_{E} \] holds for any polynomial $p$ of degree at most $n$. It admits the \textit{local Markov property} $\text{LMP}(m)$ if there exist constants $c,k\geq1$ such that \[ |p^{(j)}(z_0)|\leq\bigl(\frac{cn^k}{r^m}\Bigr)^j{\|p\|}_{E\cap{B}(z_0,r)} \] holds for all $n\in{\mathbb N}$, $j\in\{1,\ldots,n\}$, $z_0\in E$, $r\in\left(0,1\right]$, and any polynomial $p$ of degree at most $n$. Hereby, ${\|\cdot\|}_{E}$ is the supremum norm on $E$ and $B(z_0,r)$ is the closed ball with center $z_0$ and radius $r$. Note that the local Markov property implies the global Markov inequality. The main result of the paper is that $\text{GMI}(k)$ implies $\text{LMP}(m)$ (for certain $m$) if $E$ satisfies a certain \textit{Jackson property}.
reviewed by: Klaus Schiefermayr
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 31A15
Mathematics Subject Classification ID: 30A10
zbMATH DE Number: 7021558
zbMATH Keywords: Markov inequality
zbMATH Keywords: extension operators
zbMATH Keywords: Kolmogorov inequality
zbMATH Keywords: Jackson inequality
zbMATH Keywords: Green function
MaRDI profile type: Publication
full work available at URL: j.aim.2019.01.027
OpenAlex ID: W2182734994
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Wikidata QID: Q128529609
DOI: 10.1016/J.AIM.2019.01.027
description: scientific article; zbMATH DE number 7021558