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

$$\begin{aligned} D^+ := \{z \in \mathbb {C}: |z| \le 1, \text { Im}(z) \ge 0\}\,. \end{aligned}$$

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

$$\begin{aligned} \Vert f\Vert _A := \sup _{z \in A}{|f(z)|} \end{aligned}$$

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

$$\begin{aligned} c_1 \left( \frac{n}{k+1}\right) ^{1/2} \le \inf _{P}{\frac{\Vert P^{\prime }\Vert _{[-1,1]}}{\Vert P\Vert _{[-1,1]}}} \le c_2 \left( \frac{n}{k+1}\right) ^{1/2} \end{aligned}$$

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 $0 \le k \le n$.