Normal criteria on the family of meromorphic functions with shared set
Let \(\mathcal{F}\) be a family of meromorphic functions in a domain \(D\) of the extended complex plane \(\mathbb{C}^1\) and let \(S\subset V\) be a non-empty finite point set on the relevant Riemann sphere of unit diameter. Using some geometric methods and the theory of covering surfaces, the authors prove the following results about normality of \(\mathcal{F}\) with some corollaries: {Theorem 1.1} (Area criteria): Suppose that \(\mathcal{F}\) is a family of meromorphic functions in a domain \(D\subset \mathbb{C}^1\). Then the following conditions are equivalent: (i) \(\mathcal{F}\) is normal in \(D\). (ii) \(\mathcal{F}\) is equicontinuous in \(D\). (iii) For any \(\varepsilon \in (0, \pi)\) and any \(z^* \in D\), there exists a positive number \(r=r(z^*)>0\) such that, for any \(f \in \mathcal{F}\), the area of the image set \(f(\{z:|z-z^*|<r\})\) is smaller than \(\varepsilon\). {Theorem 1.2}: Suppose that \(\mathcal{F}\) is a family of meromorphic functions in a domain \(D\subset \mathbb{C}^1\) and \(S\subset V\) is a quasi-shared of \(\mathcal{F}\). If \(S\) contains at least three points, then \(\mathcal{F}\) is normal in \(D\). {Theorem 1.3}: Suppose that \(\mathcal{F}\) is a family of meromorphic functions in a domain \(D\subset \mathbb{C}^1\) and \(S\subset V\) is a shared of \(\mathcal{F}\). If \(S\) contains at least three points, then \(\mathcal{F}\) is normal in \(D\). {Theorem 1.4}: Suppose that \(\mathcal{F}\) is a family of meromorphic functions in the domain \(D\subset \mathbb{C}^1\). If \(\mathcal{F}\) has three distinct shared values, then \(\mathcal{F}\) is normal in \(D\). {Corollary 1.} Suppose that \(\mathcal{F}\) is a family of meromorphic functions in a domain \(D\subset \mathbb{C}^1\). If there exist three distinct points \(a_1, a_2, a_3 \in V\) such that, for any given \(f \in \mathcal{F}\), \(f\) omits \(a_1, a_2, a_3\) in \(D\), then \(\mathcal{F}\) is normal in \(D\).
family of meromorphic functions defined on \(D\subset \mathbb{C}^1\)
scientific article; zbMATH DE number 6108364