Annales Academić Scientiarum Fennicć
Mathematica
Volumen 38, 2013, 49-66

A CRITERION OF NORMALITY BASED ON A SINGLE HOLOMORPHIC FUNCTION II

Xiaojun Liu and Shahar Nevo

University of Shanghai for Science and Technology, Department of Mathematics
Shanghai 200093, P.R. China; Xiaojunliu2007 'at' hotmail.com

Bar-Ilan University, Department of Mathematics
52900 Ramat-Gan, Israel; nevosh 'at' macs.biu.ac.il

Abstract. 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 C. Let k \ge 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 F. Assume also that the following two conditions hold for every f \in F: (a) f(z) = 0 => f'(z) = h(z), and (b) f'(z) = h(z) => |f^(k)(z)| \le 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.

2010 Mathematics Subject Classification: Primary 30D35.

Key words: Normal family, holomorphic functions, zero points.

Reference to this article: X. Liu and S. Nevo: A criterion of normality based on a single holomorphic function II. Ann. Acad. Sci. Fenn. Math. 38 (2013), 49-66.

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doi:10.5186/aasfm.2013.3810