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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.

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