On the spherical derivative of a rational function

TL;DR

This paper investigates the spherical derivative norm of rational functions, providing estimates for individual functions and their iterates, enhancing understanding of their geometric properties.

Contribution

It introduces bounds for the spherical derivative norm of rational functions and their iterates, offering new insights into their geometric behavior.

Findings

Bounds for K(f) for individual rational functions

Estimates of K(f) for sequences of iterates

Enhanced understanding of spherical derivatives in complex dynamics

Abstract

For a rational function f we consider the norm of the derivative with respect to the spherical metric and denote by K(f) the supremum of this norm. We give estimates of this quantity K(f) both for an individual function and for sequences of iterates.