Turán-type reverse Markov inequalities for polynomials with restricted zeros (Q2049604)

Let \(\mathcal{F}_{n,k}\) be the set of all polynomials of degree at most \(n\) with complex coefficients having at least \(n-k\) zeros in \(D^+\), \(0\leq{k}\leq{n}\), where \(D^+:=\{z\in{\mathbb C}:|z|\leq1,\operatorname{Im}(z)\geq0\}\). In this paper, the author proves that there are absolute constants \(c_1,c_2>0\) such that \[ c_1\sqrt{\frac{n}{k+1}} \leq \inf_{P}\frac{\|P'\|}{\|P\|} \leq c_2\sqrt{\frac{n}{k+1}} \] holds, where the infimum is taken over all \(P\in\mathcal{F}_{n,k}\), \(P\neq0\), having at least one zero in \([-1,1]\), and \(\|\cdot\|\) is the supremum norm on \([-1,1]\).