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Uniqueness theorems for ‘meromorphic function’. (English) Zbl 1056.30031
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 M. Fang and 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 \(T(r,f)=(n+1)T(r,h)+S(r,f)\).

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory