  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.
