Application of conformal mappings to the inequalities for polynomials

Applications of the geometric theory of functions to inequalities for algebraic polynomials are considered. The main attention is paid to constructing a univalent conformal mapping for a given polynomial and to applying the Lebedev and Nehari theorems to this mapping. A new sharp inequality of Bernshtein type for polynomials with restrictions on the growth on a segment or on a circle, inequalities with restrictions on the zeros of the polynomial, and other inequalities are obtained. In particular, classical inequalities by Markov, Bernshtein, and Schur are strengthened.