On Montel's theorem and Yang's problem
The author proves the following normality criterion. Let \(\mathcal F\) be a family of functions meromorphic in the domain \(D\subset\mathbb C\), and let \(\psi(\not\equiv 0)\) be a function meromorphic in \(D\). If for every function \(f\in\mathcal F\), (1) \(f\) has only multiple zeros; (2) the poles of \(f\) have multiplicity at least 3; (3) at the common poles of \(f\) and \(\psi\), the multiplicity of \(f\) does not equal the multiplicity of \(\psi\); (4) \(f(z)\not=\psi(z)\), then \(\mathcal F\) is normal in \(D\). This gives a partial answer to a problem of L. Yang and generalizes Montel's theorem. The author also gives some examples to show the sharpness of our result. It seems that the condition (4) should be written as \(f(z)-\psi(z)\not=0\), as the the condition (4) implies \(f\) and \(\psi\) have no common pole.
scientific article; zbMATH DE number 2162035