Summary: In this paper, we continue to discuss normality based on a single holomorphic function. We obtain the following result. Let \(\mathcal F\) be a family of functions holomorphic on a domain \(D \subset \mathbb C\). Let \(k \geq 2\) be an integer and let \(h\) (\(\not\equiv 0\)) be a holomorphic function on \(D\), such that \(h(z)\) has no common zeros with any \(f\in \mathcal F\). Assume also that the following two conditions hold for every \(f \in \mathcal F\): (a) \( f (z) = 0 \Rightarrow f' (z) = h(z)\), and (b) \(f' (z) = h(z) \Rightarrow | f (k) (z)| \leq c\), where \(c\) is a constant. Then \(\mathcal F\) is normal on \(D\). A geometrical approach is used to arrive at the result that significantly improves a previous result of the authors which had already improved a result of Chang, Fang and Zalcman. We also deal with two other similar criterions of normality. Our results are shown to be sharp.
For Part I see [the author, Acta Math. Sin., Engl. Ser. 27, No. 1, 141–154 (2011; Zbl 1218.30086)].