Normality criteria of families of meromorphic functions
A family \(\mathfrak F\) of meromorphic functions defined in a domain \(G\) of the complex plane \(\mathbb C\) is said to be normal in \(G\) if for any sequence \(f_n\in\mathfrak F\) there exists a subsequence \(f_{n_j}\)such that \(f_{n_j}\) converges spherically, locally and uniformly in \(G\), to a meromorphic function or \(\infty\). The authors establish several normality criteria and prove the following main theorem: Let \(\mathfrak F\) be a family of meromorphic functions in a domain \(G\), \(a(z)\) \((\neq 0)\), \(b(z)\) and \(a_i(z)\) \((i=1,\dots,k-1)\) be holomorphic functions in \(G\), and let \(n,k\in\mathbb N\) with \(n\geq k+4\). For each \(f\in\mathfrak F\), define \[ E(f)= \Biggl\{z: f^{(k)}(z)+\sum_{i=1}^{k-1}a_i(z)f^{(i)}(z)- a(z)f^n(z)=b(z),\;z\in G\Biggr\}. \tag{1} \] If there exists a constant \(M>0\) such that, for every \(f\in\mathfrak F\) and \(z\in E(f)\), \[ |f^{(i)}(z)|\leq M,\quad i=1,\dots,k, \tag{2} \] then \(\mathfrak F\) is normal in \(G\). This should be compared with the result proved by \textit{W. Schwick} [J. Anal. Math. 52, 241-289 (1989; Zbl 0667.30028)], where the condition (1) is replaced by \[ f^{(k)}(z)+\sum_{i=1}^{k-1}a_i(z)f^{(i)}(z)-af^n(z)\neq b, \quad a, b\in \mathbb C\quad \text{with }a\neq 0 \] and (2) is omitted, which indicates that \(\mathfrak F\) is normal in \(G\).
scientific article; zbMATH DE number 1416589