A criterion of normality based on a single holomorphic function II

In this paper, we continue to discuss normality based on a single holomorphic function. We obtain the following result. Let F be a family of functions holomorphic on a domain D ⊂ C. Let k ≥ 2 be an integer and let h (⊂ 0) be a holomorphic function on D, such that h(z) has no common zeros with any f ε F. Assume also that the following two conditions hold for every f ε F: (a) f(z) = 0 → f′(z) = h(z), and (b) f′(z) = h(z) → |f^(k)| ≤ c, where c is a constant. Then 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.