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 subset of C. Let k >= 2 be an integer and let h (not equivalent to 0) be a holomorphic function on D, such that h(z) has no common zeros with any f is an element of F. Assume also that the following two conditions hold for every f is an element of F: (a) f (z) = 0 double right arrow f'(z) = h(z), and (b) f'(z) = h(z) double right arrow vertical bar f((k)) (z)vertical bar <= 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.