Weighted sharing of three values and uniqueness of meromorphic functions
Let \(f\) and \(g\) be distinct nonconstant meromorphic functions in the complex plane, \(k\) a nonnegative integer or infinity, and \(a\) a member of \(\mathbb{C}\cup \{\infty\}\). Let \(E_k(a;f)\) be the set of all \(a\)-points of \(f\) where an \(a\)-point of multiplicity \(m\) is counted \(m\) times if \(m\leq k\) and \(k+1\) times if \(m>k\). If \(E_k(a;f)= E_k(a;g)\), then \(f\) and \(g\) are said to share the value \(a\) with weight \(k\). Using weighted sharing the author proves results on the uniqueness of meromorphic functions sharing three values in the complex plane which improve theorems of \textit{H. Ueda} [Kodai Math. J. 3, 457-471 (1980; Zbl 0468.30023)], \textit{H. X. Yi} [Chin. Ann. Math., Ser. A 9, 434-439 (1988; Zbl 0699.30024)], and \textit{S. Z. Ye} [Kodai Math. J. 15, 236-243 (1992; Zbl 0767.30026)].
scientific article; zbMATH DE number 1742086