Lifts of Logarithmic Derivatives

TL;DR

This paper introduces a technique to transfer convergence properties between derivatives of meromorphic functions and applies it to demonstrate that families with bounded Schwarzian derivatives are quasi-normal.

Contribution

It provides a new method for analyzing the convergence of derivatives of meromorphic functions and establishes quasi-normality for families with bounded Schwarzian derivatives.

Findings

Transfer of convergence properties between derivatives

Families with bounded Schwarzian derivatives are quasi-normal

New technique for meromorphic function analysis

Abstract

Consider a sequence of meromorphic functions $(f_n)_n$. This paper presents a technique that enables the transfer of convergence properties from $(f_n^{(m+1)}/f_n^{(m)})_n$ to subsequences of $(f_n^{(m)}/f_n^{(m-1)})_n$. As an application, we will show that the families of functions with bounded Schwarzian derivative are quasi-normal.