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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 $$T(r,f)=(n+1)T(r,h)+S(r,f)$$.

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