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Uniqueness theorems for meromorphic functions concerning fixed-points. (English) Zbl 1067.30065
Let $$f(z)$$, $$g(z)$$ and $$\alpha(z)$$ be meromorphic functions. Assume that $$\alpha(z)$$ is a small function with respect to $$f(z)$$ and $$g(z)$$. We call $$f(z)$$ and $$g(z)$$ share the small function $$\alpha(z)$$ CM if $$f(z)- \alpha(z)$$ and $$g(z)- \alpha(z)$$ assume the same zeros with the same multiplicities. The authors consider sharing value (small function) problem in this paper for some differential polynomials in $$f(z)$$ and $$g(z)$$, namely $$F(z)= f^n(z)(f(z)- 1)f'(z)$$ and $$G(z)= g^n(z)(g(z)- 1)g'(z)$$. They note that motivations in their research are in the direction of W. K. Hayman’s question [Research problems in function theory (London: University of London, The Athlone Press) (1967; Zbl 0158.06301)], and the paper [C.-C. Yang and X. Hua, Uniqueness and value-sharing of meromorphic functions, Ann. Acad. Sci. Fenn., Math. 22, No. 2, 305–406 (1997; Zbl 0890.30019)]. In this connection, M.-L. Fang and W. Hong proved that if $$F(z)$$ and $$G(z)$$ share the value 1 CM, then $$f(z)\equiv g(z)$$ under the condition that $$f(z)$$ and $$g(z)$$ are entire and $$n\geq 11$$ [A unicity theorem for entire functions concerning differential polynomials, Indian J. Pure Appl. Math. 32, No. 9, 1343–1348 (2001; Zbl 1005.30023)].
In this paper, the authors give an improvement of the result above. They obtained that if $$F(z)$$ and $$G(z)$$ share the small function $$\alpha(z)$$ CM, then $$f(z)\equiv g(z)$$, under the condition that $$f(z)$$ and $$g(z)$$ are entire and $$n\geq 7$$. They also consider the meromorphic case. Suppose that $$n\geq 12$$ and $$F(z)$$ and $$G(z)$$ share $$z$$ CM. Then $$f(z)\equiv g(z)$$, or $g= {(n+ 2)(1- h^{n+1})\over (n+ 1)(1- h^{n+2})} \quad\text{and}\quad f= {(n+2) h(1- h^{n+1})\over (n+ 1)(1- h^{n+ 2})},$ where $$h(z)$$ is a nonconstant meromorphic function.

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