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Lifts of Logarithmic Derivatives

Matthias Grätsch

TL;DR

This work develops a lifting framework for logarithmic derivatives of meromorphic function sequences: convergence of the higher-order ratio $\frac{f_n^{(m+1)}}{f_n^{(m)}}$ to a limit induces subsequences of lower-order ratios $\frac{f_n^{(j+1)}}{f_n^{(j)}}$ that converge on large parts of the domain, with at most a one-step increase in the index of normality during lifting. Central to the method are a Zalcman-type lemma, a precise lifting lemma, and the notion of "eventually bounded" to control local behavior and ensure meromorphic limits while avoiding convergence to $\infty$. As an application, the paper proves that the family of meromorphic functions on the unit disk with bounded Schwarzian derivative is quasi-normal by first establishing normality of the pre-Schwarzian and then lifting to the derivative $f'/f$ and to $f$, leveraging the zero-free property of $f'$ when $S_f$ is holomorphic. These results extend classical normality theory for holomorphic functions to a robust, hierarchical framework for meromorphic functions with concrete implications for geometric function theory and complex dynamics.

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.

Lifts of Logarithmic Derivatives

TL;DR

This work develops a lifting framework for logarithmic derivatives of meromorphic function sequences: convergence of the higher-order ratio to a limit induces subsequences of lower-order ratios that converge on large parts of the domain, with at most a one-step increase in the index of normality during lifting. Central to the method are a Zalcman-type lemma, a precise lifting lemma, and the notion of "eventually bounded" to control local behavior and ensure meromorphic limits while avoiding convergence to . As an application, the paper proves that the family of meromorphic functions on the unit disk with bounded Schwarzian derivative is quasi-normal by first establishing normality of the pre-Schwarzian and then lifting to the derivative and to , leveraging the zero-free property of when is holomorphic. These results extend classical normality theory for holomorphic functions to a robust, hierarchical framework for meromorphic functions with concrete implications for geometric function theory and complex dynamics.

Abstract

Consider a sequence of meromorphic functions . This paper presents a technique that enables the transfer of convergence properties from to subsequences of . As an application, we will show that the families of functions with bounded Schwarzian derivative are quasi-normal.
Paper Structure (3 sections, 7 theorems, 15 equations)

This paper contains 3 sections, 7 theorems, 15 equations.

Table of Contents

  1. Introduction and Statement of Results
  2. Auxiliary Lemmas and their Proofs
  3. Proofs of the Main Results

Key Result

Theorem 1.2

Let $D\subseteq\mathbb{C}$ be a domain, $m\in\mathbb{N}$ and $(f_n)_n\subseteq\mathcal{M}(D)$, such that $(f_n^{(m+1)}/f_n^{(m)})_n$ converges locally uniformly to some $F_m\in\mathcal{M}(D)$. Then we get for $0\leq j\leq m$ that $(f_n^{(j+1)}/f_n^{(j)})_n$ is $Q_{(m-j)}$-normal. In particular, ther

Theorems & Definitions (11)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Lemma 2.1: Zalcman, Zalcman's Lemma
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 1 more