Turán-type reverse Markov inequalities for polynomials with restricted zeros

Abstract

Let ${\mathcal{P}}_n^c$ denote the set of all algebraic polynomials of degree at most $n$ with complex coefficients. Let $$D^{+}:=\{z\in \mathbb{C}:|z|\le 1,\Im (z)\ge 0\}$$ be the closed upper half-disk of the complex plane. For integers $0\le k\le n$ let ${\mathcal{F}}_{n,k}^c$ be the set of all polynomials $P\in {\mathcal{P}}_n^c$ having at least $n-k$ zeros in $D^{+}$. Let $$\Vert f{\Vert }_A:=\underset{z\in A}{\sup }|f(z)|$$ for complex-valued functions defined on $A\subset \mathbb{C}$. We prove that there are absolute constants $c_1>0$ and $c_2>0$ such that $$c_1{\left(\frac{n}{k+1}\right)}^{1/2}\le \underset{P}{\inf }\frac{\Vert P^{\prime }{\Vert }_{[-1,1]}}{\Vert P{\Vert }_{[-1,1]}}\le c_2{\left(\frac{n}{k+1}\right)}^{1/2}$$ for all integers $0\le k\le n$, where the infimum is taken for all $0\not\equiv P\in {\mathcal{F}}_{n,k}^c$ having at least one zero in $[-1,1]$. This is an essentially sharp reverse Markov-type inequality for the classes ${\mathcal{F}}_{n,k}^c$ extending earlier results of Turán and Komarov from the case $k=0$ to the cases $0\le k\le n$.