Tur\'an-type reverse Markov inequalities for polynomials with restricted zeros

Abstract

Let ${\cal P}_n^c$ denote the set of all algebraic polynomials of degree at most $n$ with complex coefficients. Let $$D^+ := \{z \in \mathbb{C}: |z| \leq 1, \, \, \Im(z) \geq 0\}$$ be the closed upper half-disk of the complex plane. For integers $0 \leq k \leq n$ 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 $$\|f\|_A := \sup_{z \in A}{|f(z)|}$$ for complex-valued functions defined on $A \subset {\Bbb C}$. We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 \left(\frac{n}{k+1}\right)^{1/2} \leq \inf_{P}{\frac{\|P^{\prime}\|_{[-1,1]}}{\|P\|_{[-1,1]}}} \leq c_2 \left(\frac{n}{k+1}\right)^{1/2}$$ for all integers $0 \leq k \leq n$, 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 Tur\'an and Komarov from the case $k=0$ to the cases $0 \leq k \leq n$.