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

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.


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


Zbl 0758.30028
Full Text: DOI


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