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Normality and shared values. (English) Zbl 1079.30044
For $f$ meromorphic on the unit disc ${\Delta }$ and $a\in ℂ$ define ${\overline{E}}_{f}\left(a\right)={f}^{-1}\left(\left\{a\right\}\right)\cap {\Delta }=\left\{\phantom{\rule{0.166667em}{0ex}}z\in {\Delta }:f\left(z\right)=a\phantom{\rule{0.166667em}{0ex}}\right\}$. Two functions $f$ and $g$ on ${\Delta }$ are said to share the value $a$ if ${\overline{E}}_{f}\left(a\right)={\overline{E}}_{g}\left(a\right)$. A meromorphic function $f$ on $ℂ$ is called a normal function if there exists a positive number $M$ such that ${f}^{#}\left(z\right)\le M$, where ${f}^{#}\left(z\right)=|{f}^{\text{'}}{\left(z\right)|/\left(1+|f\left(z\right)|}^{2}\right)$ denotes the spherical derivative. The authors prove the following theorems: 1) Let $ℱ$ 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 $f\in ℱ$, ${\overline{E}}_{f}\left(0\right)={\overline{E}}_{{f}^{\text{'}}}\left(a\right)$, ${\overline{E}}_{f}\left(c\right)={\overline{E}}_{{f}^{\text{'}}}\left(b\right)$, then $ℱ$ 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 $ℂ$ and $a$ and $b$ be distinct complex numbers. If $f$ and ${f}^{\text{'}}$ 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 Bloch functions, normal functions, normal families
##### Keywords:
normal families; shared values
##### References:
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