Creating Limit Functions By The Pang-Zalcman Lemma

TL;DR

This paper extends Zalcman's Lemma to analyze the limit functions of a family of functions involving rational and polynomial components, providing new insights into their behavior in complex analysis.

Contribution

It introduces a method to compute limit functions for a family of functions using an extension of Zalcman's Lemma, focusing on functions of the form Re^P.

Findings

Derived explicit forms of limit functions for the family {f(nz)}

Extended Zalcman's Lemma to a broader class of functions

Provided new tools for analyzing non-normal families in complex analysis

Abstract

In this paper we calculate the collection of limit functions obtained by applying an extension of Zalcman's Lemma, due to X. C. Pang, to the non-normal family $\left\{f(nz):n\in\mathbb{N}\right\}$ in $\mathbb{C}$, where $f=Re^P$. Here $R$ and $P$ are an arbitrary rational function and a polynomial, respectively, where $P$ is a non-constant polnomial.