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.

MSC:

 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:

References:

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