On Turán type inequalities related to metric estimates of simple partial fractions

In 1939 P. Turán established the following inequality for the derivative $P_n'$ of algebraic polynomials $P_n$ of degree $n$, all of whose zeros lie in the closed unit interval $-1\le x\le 1$,
$$\|P_n'\|_{C[-1,1]}>\frac{\sqrt{n}}{6}\,\|P_n\|_{C[-1,1]},$$
converse to the classical A. A. Markov inequality (which is true for arbitrary polynomials). In the talk, we discuss a number of generalizations of Turán's inequality, in particular, the case when the zeros of a polynomial are taken on a set larger than the unit interval. The technique of constructing these generalizations uses the apparatus of metric estimates of simple partial fractions (that is, the logarithmic derivatives $\rho_n(x)=P_n'(x)/P_n(x)=\sum_{k=1}^n (x-z_k)^{-1}$ of polynomials $P_n$). By metric estimates of simple partial fractions, we mean estimates of the measure of sets of the form
$$\{x\in E: \|\rho_n(x)|\ge \delta\}, \qquad \delta>0$$
(probably, with some non-negative weight function near the simple partial fraction $\rho_n$), under certain restrictions on the poles $z_k$ of the fraction; here $E$ is a fixed subset of a real line (usually, $E=[-1,1]$ or $E=\mathbb{R}$).