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)].


30D45 Normal functions of one complex variable, normal families
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