Equivalence of the global and local Markov inequalities in the complex plane (Q1721968)

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}.