Four-point distortion theorem for complex polynomials

TL;DR: In this article, a lower bound on distortion of cross ratio of four points under the mapping effected by a complex polynomial with restricted critical values was established, and an exact lower bound for maximal moduli of critical values of polynomials of degree n was established.

Abstract: We prove a theorem on distortion of cross ratio of four points under the mapping effected by a complex polynomial with restricted critical values. Its corollaries include inequalities involving the absolute value and certain coefficients of a polynomial. In particular, an exact lower bound is established for maximal moduli of critical values of polynomials $P$ of degree $n$ normalized by $P(0)=0$, $P'(0)\ne0$.