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Uniqueness of meromorphic functions that share three values. (English) Zbl 0767.30026
The author proves the following theorem on shared values of meromorphic functions. Theorem. Let \(f\) and \(g\) be two distinct nonconstant meromorphic functions such that \(f\) and \(g\) share 0, 1, \(\infty\) with the same multiplicities (i.e. C.M.), and let \(a_ 1,a_ 2,\dots,a_ p\) be \(p\) (\(\geq 1\)) distinct finite complex numbers, and \(a_ i\neq 0\) (\(i=1,2,\dots,p\)). If \[ \sum_{i=1}^ p f(a_ i,f)+f(\infty,f)>{{2(p+1)} \over {p+2}}, \] then there exists one and only one \(a_ j\) in \(a_ 1,a_ 2,\dots,a_ p\) such that \(a_ j\) and \(1- a_ j\) are Picard exceptional values of \(f\) and \(g\) respectively, and also \(\infty\) is so, and \[ (f-a_ j)(g+a_ j-1)\equiv a_ j(1-a_ j). \] {}.

MSC:
30D30 Meromorphic functions of one complex variable (general theory)
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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