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 Turan and Komarov from the case $$k=0$$ to the cases $$0 \le k \le n$$.