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

Mikhail A. Komarov

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

Mikhail A. Komarov

Abstract

Let $K\subset \mathbb{C}$ be a convex compact set, and let $Π_n(K)$ be the class of polynomials of exact degree $n$, all of whose zeros lie in $K$. The Turán type inverse Markov factor is defined by $M_n(K)=\inf_{P\in Π_n(K)} \left(\|P'\|_{C(K)}/\|P\|_{C(K)}\right)$. A combination of two well-known results due to Levenberg and Poletsky (2002) and Révész (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 $n,w,d$.

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

Abstract

Let

be a convex compact set, and let

be the class of polynomials of exact degree

, all of whose zeros lie in

. The Turán type inverse Markov factor is defined by

. A combination of two well-known results due to Levenberg and Poletsky (2002) and Révész (2006) provides the lower bound

, where

is the diameter of

and

is the minimal width (the smallest distance between two parallel lines between which

lies). We prove that this bound is essentially sharp, namely,

for all

Paper Structure (3 theorems, 72 equations)

This paper contains 3 theorems, 72 equations.

Key Result

Theorem 1

There are absolute constants $c_1,c_2>0$ such that for any convex compact set $K$ and $n\ge 1$ we have where $w=w(K)$, $d=d(K)$ and the value $c_2:=28$ is suitable. The upper bound in $(main th_0)$ is true for any connected compact set $K$.

Theorems & Definitions (3)

Theorem 1
Corollary 1
Corollary 2