Normal families of meromorphic functions concerning shared values. (English) Zbl 1145.30013

Zhang, Qingcai

J. Math. Anal. Appl. 338, No. 1, 545-551 (2008).

Let \({\mathcal F}\) be a family of meromorphic functions in a domain \(D\). It is known that if every function in \({\mathcal F}\) omits three distinct values, then \({\mathcal F}\) is normal. W. Schwick [Arch. Math. 59, No. 1, 50–54 (1992; Zbl 0758.30028)] obtained a normality criteria from the point of view of value distribution theory, in particular, shared values. The author considers the sharing conditions with differential polynomials. Let \(n\) be a positive integer, and \(a\) be a nonzero constant. If \(n\geq 4\) and for each pair of \(f\) and \(g\) in \({\mathcal F}\), \(f'- af^n\) and \(g'- ag^n\) share a value \(b\), then \({\mathcal F}\) is normal. The author also considers a family of entire functions. Examples are given which imply that results in this paper are sharp. The methods for the proofs are the value distribution theory and Zalcman’s lemma.

Reviewer: Katsuya Ishizaki (Saitama)

MSC:

30D35	Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D45	Normal functions of one complex variable, normal families

Keywords:

differential polynomial; Zalcman’s lemma

Citations:

Zbl 0758.30028

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