On the precise form of the inverse Markov factor for convex sets

TL;DR

This paper establishes that the inverse Markov factor for convex sets in the complex plane is tightly bounded, providing a nearly optimal upper bound that matches known lower bounds up to a constant factor.

Contribution

It proves that the known lower bound for the inverse Markov factor is essentially sharp by establishing a matching upper bound with a universal constant.

Findings

The inverse Markov factor is bounded above by a constant multiple of the known lower bound.

The bounds depend explicitly on the convex set's diameter and minimal width.

The results confirm the sharpness of previous lower bounds for convex sets in complex analysis.

Abstract

Let $K\subset \mathbb{C}$ be a convex compact set, and let $\Pi_n(K)$ be the class of polynomials of exact degree $n$, all of whose zeros lie in $K$. The Tur\'an type inverse Markov factor is defined by $M_n(K)=\inf_{P\in \Pi_n(K)} \left(\|P'\|_{C(K)}/\|P\|_{C(K)}\right)$. A combination of two well-known results due to Levenberg and Poletsky (2002) and R\'ev\'esz (2006) provides the lower bound $M_n(K)\ge c\left(wn/d^2+\sqrt{n}/d\right)$, $c:=0.00015$, where $d>0$ is the diameter of $K$ and $w\ge 0$ is the minimal width (the smallest distance between two parallel lines between which $K$ lies). We prove that this bound is essentially sharp, namely, $M_n(K)\le 28\left(wn/d^2+\sqrt{n}/d\right)$ for all $n,w,d$.