## Normality and shared values.(English)Zbl 1079.30044

For $$f$$ meromorphic on the unit disc $$\Delta$$ and $$a\in \mathbb{C}$$ define $$\overline E_f(a)=f^{-1}(\{a\})\cap\Delta=\{\,z\in\Delta:f(z)=a\,\}$$. Two functions $$f$$ and $$g$$ on $$\Delta$$ are said to share the value $$a$$ if $$\overline E_f(a)=\overline E_g(a)$$. A meromorphic function $$f$$ on $$\mathbb{C}$$ is called a normal function if there exists a positive number $$M$$ such that $$f^\#(z)\leq M$$, where $$f^\#(z)=| f'(z)| /(1+| f(z)| ^2)$$ denotes the spherical derivative. The authors prove the following theorems: 1) Let $$\mathcal{F}$$ be a family of meromorphic functions on the unit disc $$\Delta$$, and let $$a$$ and $$b$$ be distinct complex numbers and $$c$$ a nonzero complex number. If for every $$f\in \mathcal{F}$$, $$\overline E_f(0)=\overline E_{f'}(a)$$, $$\overline E_f(c)=\overline E_{f'}(b)$$, then $$\mathcal{F}$$ is normal on $$\Delta$$. Earlier a similar result has been proved by W. Schwick [Arch. Math. 59, No. 1, 50–54 (1992; Zbl 0758.30028)]. 2) Let $$f$$ be a meromorphic function on $$\mathbb{C}$$ and $$a$$ and $$b$$ be distinct complex numbers. If $$f$$ and $$f'$$ share $$a$$ and $$b$$, then $$f$$ is a normal function. This should be compared with the result of E. Mues and N. Steinmetz in [Manuscr. Math. 29, 195–206 (1979; Zbl 0416.30028)].

### MSC:

 30D45 Normal functions of one complex variable, normal families

### Keywords:

normal families; shared values

### Citations:

Zbl 0758.30028; Zbl 0416.30028
Full Text:

### References:

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