Multi-point inequalities for derivatives of rational functions accounting for the configuration of their zeros

Abstract

This paper investigates the geometric properties of rational functions by establishing new multi-point inequalities for their derivatives. The study focuses on a class of rational functions of degree \(n \geqslant 2\) whose critical values are concentrated on a given segment [a, b] of the real axis. An approach based on the metric characteristics of the mutual arrangement of zeros (or poles) is used. This method utilizes the concepts of inner radii and the Green function for multiply connected domains to refine existing distortion theorems. Several statements are proven to provide estimates for the product of derivative moduli at the functions’ zeros located on the complex plane.