Uniqueness of entire functions that share some small functions
As usual in the Nevanlinna theory of meromorphic functions, let \(S(r,f)\) denote any quantity such that \(S(r,f)=o\bigl(T(r,f)\bigr)\) as \(r\to\infty\), outside of a possible exceptional set of \(r\)-values of finite linear measure. A meromorphic function~\(\alpha\) is called to be small with respect to~\(f\) whenever \(T(r,\alpha)=S(r,f)\). Recall that two meromorphic functions \(f\), \(g\) share a constant \(\alpha\in\hat{\mathbb C}\) or a meromorphic function \(\alpha(z)\) IM, resp. CM, if \(f-\alpha\) and \(g-\alpha\) have the same zeros ignoring multiplicities, resp. counting multiplicities. The following theorem is proved in this article: Let \(f\) be nonconstant entire, and let \(a\), \(b\) be two distinct meromorphic functions small with respect to~\(f\). If \(f\) and \(f'\) share \(a\) and~\(b\) IM, then \(f=f'\). This generalizes earlier results due to \textit{L.~Rubel} and \textit{C. C.~Yang} [Lect. Notes Math. 599, 101--103 (1977; Zbl 0362.30026)], \textit{E.~Mues} and \textit{N. Steinmetz} [Manuscr. Math. 29, 195--206 (1979; Zbl 0416.30028)] and \textit{J. Zheng} and \textit{S. Wang} [Adv. Math. (China) 21, 334--341 (1992; Zbl 0783.30026)].
scientific article; zbMATH DE number 1496080