Normal families and uniformly discrete exceptional sets (Q2166133)

Let \(\mathcal{F}\) be a family of meromorphic functions defined in a domain \(D\). In the paper the authors investigate the properties of the family \(\mathcal{F}\) near its non-normal points. For \(f \in \mathcal{F}\), let us define a set \(E_{f}\) as follows: \(E_{f} = \{ z \in D: f(z) = 0\} \cup \{z \in D: f^{(k)}(z) = h(z)\}\) for a given function \(h\) defined in \(D\). \par The family \(\{E_{f} : f \in \mathcal{F}\}\) is called locally uniformly discrete in \(D\), if for each point \(z_{0} \in D\), there exists a \(\delta(>0)\) such that for each \(E_{f}\), \(E_{f} \cap \Delta_{\delta}(z_{0})\) contains at most one point, where \(\Delta_{\delta}(z_{0}) = \{z \in D: \mid z - z_{0} \mid < \delta\}\). \par The main result of the paper is as follows: Let \(k\) be a positive integer and \(h (\not\equiv 0)\) be a holomorphic function in a domain \(D\) and let \(\mathcal{F}\) be a family of meromorphic functions defined in \(D\), all of whose zeros have multiplicities at least \(k + 2\). Suppose that at each common zero of \(f \in \mathcal{F}\) and \(h\), the multiplicities \(m_{f}\) for \(f\) and \(m_{h}\) for \(h\) satisfy \(m_{f} \geq m_{h} + k + 1\) for \(k > 1\) and \(m_{f} \geq 2m_{h} + 3\) for \(k = 1\), and the family of sets \(\{E_{f}: f \in \mathcal{F}\}\) is locally uniformly discrete. If \(\mathcal{F}\) is not normal at \(z_{0} \in D\), then \(z_{0}\) is a simple zero of \(h\), and there exists a \(\delta > 0\) and \(\{f_{n}\} \subset \mathcal{F}\) such that \[f_{n}(z) = \frac{(z - z_{n,0})^{k + 2}}{z - z_{n, \infty}}\hat{f}_{n}(z)\] on \(\Delta_{\delta}(z_{0})\), where \(\displaystyle \frac{z_{n, 0} - z_{0}}{\rho_{n}} \to c\) and \(\displaystyle \frac{z_{n, \infty} - z_{0}}{\rho_{n}} \to (k + 2)c\) for some sequence of positive numbers \(\rho_{n} \to 0\) and constant \(c \neq 0\). Moreover, \(\hat{f}_{n}(z)\) has no zero and is holomorphic on \(\Delta_{\delta}(z_{0})\), where \(\hat{f}\) satisfies \(\displaystyle ((z - z_{0})^{k + 1}\hat{f}(z))^{(k)} \equiv h(z)\).