A criterion of normality based on a single holomorphic function II

TL;DR

This paper establishes new criteria for the normality of families of holomorphic functions based on a single holomorphic function, improving previous results with a geometric approach and demonstrating the sharpness of these criteria.

Contribution

It introduces novel normality criteria involving a holomorphic function with no common zeros, extending and sharpening earlier results through a geometric method.

Findings

The criteria are sharp and cannot be improved.

Normality is guaranteed under specified derivative conditions.

The approach improves previous normality criteria in complex analysis.

Abstract

In this paper, we continue to discuss normality based on a single\linebreak holomorphic function. We obtain the following result. Let $\CF$ be a family of functions holomorphic on a domain $D\subset\mathbb C$. Let $k\ge2$ be an integer and let $h(\not\equiv0)$ be a holomorphic function on $D$, such that $h(z)$ has no common zeros with any $f\in\CF$. Assume also that the following two conditions hold for every $f\in\CF$:\linebreak %{enumerate} [(a)] (a) $f(z)=0\Longrightarrow f'(z)=h(z)$ and %[(b)] (b) $f'(z)=h(z)\Longrightarrow|f^{(k)}(z)|\le c$, where $c$ is a constant. Then $\CF$ is normal on $D$. %{enumerate} A geometrical approach is used to arrive at the result which significantly improves the previous results of the authors, \textit{A criterion of normality based on a single holomorphic function}, Acta Math. Sinica, English Series (1) \textbf{27} (2011), 141--154 and of Chang, Fang, and Zalcman, \textit{Normal families of holomorphic functions}, Illinois Math. J. (1) \textbf{48} (2004), 319--337. We also deal with two other similar criterions of normality. Our results are shown to be sharp.