Value distribution of meromorphic function concerning shared-values
Let \(D\) be a domain in the complex plane \(\mathbb C\) and let \(f(z)\) be a meromorphic function in \(D\). For \(a\in \mathbb C\), the authors set \(\overline E(a, f)= f^{-1}(\{a\})\cup D= \{z\in D: f(z) = a\}\). They call two meromorphic functions \(f(z)\) and \(g(z)\) share value \(a\) on \(D\) if \(\overline E(a, f) =\overline E(a, g)\). The authors consider value distribution problems under some sharing value conditions on meromorphic functions and its derivatives in the whole complex plane \(\mathbb C\) and in the unit disk \(\Delta\). The result for meromorphic function \(f\) in \(\mathbb C\) is that if \(f\) and \(f'\) share \(a\in \mathbb C\), then \(f'\) assumes every finite value infinitely often except possibly one value \(b\), where \(b\in \{0, a/(k + 1)\}\), \(k\) is a positive integer. Let \(F\) be a family of meromorphic functions on \(\Delta\). It is shown that if there exist finite numbers \(a\) and \(b\) (\(b\neq 0\), \(a/b\) is not any positive integer) such that for any \(f\in F\), \(f\) and \(f'\) share \(a\) in \(\Delta\) and \(| f(z)- a|\geq\varepsilon\) holds if \(f'(z)= b\) where \(\varepsilon\) is a positive number, then \(F\) is normal on \(\Delta\). This result connects with [\textit{W. Schwick}, Arch. Math. 59, No. 1, 50--54 (1992; Zbl 0758.30028)]. The main tools for the proofs is Zalcman's rescaling lemma, e.g. [\textit{L. Zalcman}, Bull. Am. Math. Soc., New Ser. 35, No. 3, 215--230 (1998; Zbl 1037.30021)].
scientific article; zbMATH DE number 2024723