Two new criteria for normal families
The author considers a family \(\mathbf F\) of meromorphic functions defined on the domain \(\Omega\) and shows when the set \(\{f'/f \in \mathbf F\}\) is a normal family. The following theorems are the main results of this paper. Theorem 1. Let \(k \geq 2\) and \(a_0, \ldots, a_{k-1}\) be analytic functions on \(\Omega.\) If \(f\) and \(L(f):= f^{(k)} + a_{k-1} f^{(k-1)} + \ldots + a_1 f' + a_0 f\) are the functions without zeros on \(\Omega\) for each \(f \in \mathbf F\) then \(\{f'/f \in \mathbf F\}\) is a normal family on \(\Omega.\) Theorem 2. Let \(k \geq 2\) and \(a_0, \ldots, a_{k-1}\) be meromorphic functions on \(\Omega.\) If \(f,\) \(f'\) and \(L(f)\) are the functions without zeros on \(\Omega\) for each \(f \in \mathbf F\) then \(\{f'/f \in \mathbf F\}\) is a normal family on \(\Omega.\)
scientific article; zbMATH DE number 5011336
scientific article