Normality criteria for families of meromorphic algebroid functions (Q622498)

Giving subsets \(A\) and \(B\) of \(\mathbb{C}_{\infty}\), the authors set \(H(A,B)\) the infimum of the \(\delta>0\) such that each of \(A,B\) lies inside a \(\delta\)-neighbourhood (measured by the sphere distance) of the other. That is \[ H(A,B)=\inf\{\delta>0: A\subset B_{\delta)},B\subset A_{\delta)}\}, \] here \(A_{\delta)}=\{z\in\mathbb{C}_{\infty}:|z,A|<\delta\},|z,A|=\inf\{|z,a|;a\in A\}\), where \(|z,a|\) is the sphere distance of \(z\) and \(a\). By the Hausdorff distance \(H(A,B)\), the authors prove some normality criteria for families of algebroidal functions. One of the main results is: Suppose the family of \(v\)-valued algebroidal functions \(\mathcal F\) has no branch points in a simply connected domain \(D\). If there exist mutually distinct points \(b_{1}\), \(b_{2}\) and \(b_{3}\) on the Riemann sphere such that the range of any \(F\in\mathcal F\) does not include them (\(F(D)\cap\{b_{j}\}_{j=1}^3\neq\emptyset\)), then \(\mathcal F\) is a normal family on \(D\). And for every sequence \(\{F_{n}\}\subset\mathcal F\), there is a subsequence which is convergent uniformly on closed subsets of \(D\) to a algebraic function \(F_0\) with respect to Hausdorff distance \[ F_{n_{t}}(Z)\rightarrow F_{0}(z)\quad(t\rightarrow \infty). \] Finally, some examples are also given to support the theories in this paper.