Normal families and shared values of holomorphic functions
Let \(\mathbb C\) be the complex plane, and let \(f\) be a meromorphic function in a domain \(D\subset\mathbb C\). For \(a\in\mathbb C \), set \({\overline E}_f(a)=\{\,z\in\mathcal D:f(z)=a\,\}\). Let \(\mathcal F\) be a family of meromorphic functions in a domain \(\mathcal D\subset\mathbb C\). Then \(\mathcal F\) is said to be normal in \(\mathcal D\) if every sequence \(\{f_n\}\subseteq\mathcal F\) contains a subsequence which converges spherically uniformly on compact subsets of \(\mathcal D\). The main theorem in the paper under review is the following: Let \(\mathcal F\) be a family of holomorphic functions in a domain \(\mathcal D\), let \(k\) be a positive integer, \(a\), \(b(\neq 0)\), \(c(\neq0)\) and \(d\) be finite complex numbers. If, for each \(f\in\mathcal F\), all zeros of \(f(z)-d\) have multiplicity at least \(k\), \({\overline E}_f(0)\subseteq{\overline E}_{f^{(k)}}(a)\), and \({\overline E}_{f^{(k)}}(b)\subseteq{\overline E}_g(c)\), then \(\mathcal F\) is normal in \(\mathcal D\). This result extends the well-known normality criterion of C. Miranda and improves some results due to \textit{Y. Xu} [Indian J. Pure App. Math., 30, No. 3, 287--293 (1999; Zbl 0922.30028)], \textit{H. H. Chen} and \textit{M. L. Fang} [J. Math. Anal. Appl., 260, No. 1, 124--132 (2001; Zbl 1020.30035)], \textit{X. C. Pang} [Analysis 22, No. 2, 175--182 (2002; Zbl 1030.30030] and \textit{M. L. Fang} and \textit{Y. Xu} [Israel J. Math. 129, 125--141 (2002; Zbl 1007.30025)]. Further, some examples are provided to show that the obtained results are sharp.
scientific article; zbMATH DE number 5077209