A criterion of normality based on a single holomorphic function II

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.