Uniqueness theorems for `meromorphic function' (Q1431052)

This paper is considering uniqueness results of meromorphic functions under suitable value sharing conditions for their differential polynomials. The main result, improving a result by \textit{M. Fang} and \textit{W. Hong}, see [Indian J. Pure Appl. Math. 32, 1343--1348 (2001; Zbl 1005.30023)], reads as follows: Given two transcendental entire functions \(f\) and \(g\) and an integer \(n \geq 7\), if \(f^n(f-1)f'\) and \(g^n(g-1)g'\) share the value one CM, then \(f=g\). The previous result had \(n \geq 11\). Several related results will be proved as well, including the following meromorphic variant of the main result: Given two distinct nonconstant meromorphic functions \(f\) and \(g\) and an integer \(n \geq 12\), then under the same shared value condition, there exists a nonconstant meromorphic function \(h\) so that \[ g = \frac{(n+2)(1-h^{n+1})}{(n+1)(1-h^{n+2})}, \enspace f=hg. \] The proofs rely on the Nevanlinna theory. Some details remained unclear for the reviewer, say e.g.\ the proof of Theorem 2, where clearly \(T(r,f) = (n+2)T(r,h)+S(r,f)\) should have been written instead of \(T(r,f)=(n+1)T(r,h)+S(r,f)\).