On the distortion theorems for algebraic polynomials

The applications of a boundary Schwarz lemma and the properties of the condenser capacity to some inequalities for polynomials and their derivatives are considered. We prove a new Bernstein-type inequality for the polynomials on a circle, two-sided estimates for the polynomials with constraints on their critical values, and two-sided estimates of the average distortion computed at zeros of the polynomials.