Normality and shared-value of derivatives

Abstract

We show that a family F of functions holomorphic in a domain is normal, if for all ƒ ∈ F the derivative ƒ′ and ƒ share the value a(≠0) and if the set of values that ƒ _(k+1) takes the zeros of ƒ−a, is bounded. As an application we obtain some new uniqueness theorems of entire functions which improve the former results [Amer. H.H. Al-khaladi (2000). On a question of Yi-Yang. J. Shandong Univ., 35(3), 162–167; G. Jank, E. Mues and L. Volkmann (1986). Meromorphic functionen die mit ihrer ersten und zweiten Ableitung einen endlichen wert teilen. Complex Variables, 6, 51–71; P. Li and C.C. Yang (2001). Uniqueness theorem on entire functions and their derivatives. J. Math. Anal. Appl., 253, 50–57; W. Lin (2001). Uniqueness theorem and normal criteria with sharing array. Math. Theory Appl., 21, 9–16 (Chinese); H.X. Yi and C.C. Yang (1995). Uniqueness Theory of Meromorphic Functions, Pure and Applied Math. Monographs No. 32. Science Press, Beiling; H. Zhong (1995). Entire functions that share one values with their derivatives. Kodai Math. J., 18, 250–259, etc].

Keywords:

• Holomorphic functions
• Normal families
• Normal functions
• Shared-value
• AMS Subject Classifications: 30D35
• 30D45

Acknowledgement

The authors would like to thank the referee for his/her valuable comments. Project supported by Shandong Province and National Natural Science Foundation of China.