Four-point distortion theorem for complex polynomials

TL;DR

This paper establishes a four-point distortion theorem for complex polynomials with restricted critical values, deriving inequalities and bounds related to polynomial coefficients and critical moduli.

Contribution

It introduces a new distortion theorem for four points under polynomial mappings with restricted critical values, providing exact bounds and inequalities.

Findings

Proves a distortion theorem for four points under such polynomials

Derives inequalities involving polynomial coefficients and absolute values

Establishes an exact lower bound for maximal critical value moduli

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 $P(0)=0$, $P'(0)\ne0$.