## A note on normality and shared values.(English)Zbl 1074.30032

Let $$D$$ be a domain in the complex plane. For a complex value $$b$$ in the extended complex plane, two functions $$f$$ and $$g$$ meromorphic in $$D$$ are said to share the value $$b$$ if $$f(z) = b$$ if and only if $$g(z) = b$$. The authors prove some sufficient conditions for a family $${\mathcal F}$$ of functions meromorphic in $$D$$ to be a normal family in $$D$$.
Theorem 1: If $$k$$ is a positive integer and $${\mathcal F}$$ is a family of functions meromorphic in $$D$$ such that each zero of each function $$f \in {\mathcal F}$$ has multiplicity at least $$k + 2$$, and if, for each pair of functions $$f, g \in {\mathcal F}$$, both $$(i)$$ $$f$$ and $$g$$ share the value $$0$$, and $$(ii)$$ $$f^{(k)}$$ and $$g^{(k)}$$ share a fixed value $$b$$, then $${\mathcal F}$$ is a normal family in $$D$$.
Theorem 2: If $$k$$ is a positive integer and $${\mathcal F}$$ is a family of functions meromorphic in $$D$$ such that each zero of each function $$f \in {\mathcal F}$$ has multiplicity at least $$k + 1$$, each pole of each function $$f \in {\mathcal F}$$ has multiplicity at least 2, and if, for each pair of functions $$f, g \in {\mathcal F}$$, both $$(i)$$ $$f$$ and $$g$$ share the value 0, and $$(ii')$$ $$f^{(k)}$$ and $$g^{(k)}$$ share a fixed value $$b$$, then $${\mathcal F}$$ is a normal family in $$D$$.
Since functions holmorphic in $$D$$ lack poles, Theorem 2 applies to a family of holomorphic functions. As a corollary, it is shown that if m is a positive integer and $${\mathcal F}$$ is a family of functions meromorphic in $$D$$ such that for each pair of functions $$f, g \in {\mathcal F}$$, both $$(i)$$ $$f$$ and $$g$$ share the value 0, and $$(ii''$$) $$f^{m+1} f'$$ and $$g^{m+1} g'$$ share a fixed value $$b$$, then $${\mathcal F}$$ is a normal family in $$D$$. An example is given showing that the order of the derivatives in Theorem 1 cannot be reduced. The proofs of the theorems involve a lengthy elimination of cases, and use some results from Nevanlinna theory. The case of Theorem 1 where each $$f \in {\mathcal F}$$ omits the value 0 and each $$k$$-th derivative of $$f$$ omits a fixed value $$b$$ is due to Y. X. Gu [Sci. Sinica, Special Issue 1, 267–274 (1979)].

### MSC:

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

### Keywords:

shared values; normal family
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### References:

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