Normal families and shared values of meromorphic functions. (English) Zbl 1179.30033

Let \(f\) and \(g\) be meromorphic functions on a domain \(D\), and let \(a\) and \(b\) be two complex numbers. If \(g(z)=b\) whenever \(f(z)=a\), we write \(f(z)=a\Rightarrow g(z)=b\). If \(f(z)=a\Rightarrow g(z)=b\) and \(g(z)=b\Rightarrow f(z)=a\), we write \(f(z)=a\Leftrightarrow g(z)=b\).
The authors of this paper prove the following theorem which improves a result of M. Fang and L. Zalcman [Comput. Methods Funct. Theory 2, No. 2, 385–395 (2002; Zbl 1048.30018)].
Let \(\mathcal F\) be a family of meromorphic functions in a domain \(D\), let \(q, k\) be two positive integers, and let \(a, b\) be two non-zero complex numbers. If, for each \(f\in \mathcal F\), the zeros of \(f\) have multiplicity at least \(k+1\), and \(f=a\Leftrightarrow G(f)=b\), where \(G(f)=P(f^{(k)})+H(f)\) is a differential polynormial of \(f\) satisfying \(q\geq \gamma_H\) and \(\frac{\Gamma}{\gamma}|_{H}<k+1\), then \(\mathcal F\) is normal in \(D\).


30D45 Normal functions of one complex variable, normal families


Zbl 1048.30018
Full Text: DOI Euclid


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