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Unicity theorems for meromorphic functions that share three values. (English) Zbl 0868.30032
Two meromorphic functions \(f\) and \(g\) share the complex value \(a\) if \(f(z)=a\) implies \(g(z)=a\) and vice versa. The value \(a\) is shared CM (counting multiplicities) if, in addition, \(f\) and \(g\) have the same multiplicities at each \(a\)-point.
Let \(f\) and \(g\) be distinct non-constant meromorphic functions in the complex plane sharing the value \(0\), \(1\) and \(\infty\) CM.
The main result of the paper under review is Theorem 4: If \[ N(r,1/(f-a))\neq T(r,f)+ S(r,f)\tag{\(*\)} \] then \(f\) and \(g\) are related by one of the following relations: \((f-a)(g+a-1)= a(1-a)\), \(f+(a-1)g=a\) or \(f=ag\).
G. Brosch [Eindeutigkeitssätze für meromorphe Funktionen, Dissertation, RWTH Aachen (1989; Zbl 0694.30027)] already proved (Folgerung 5.2, which is a corollary to a result of E. Mues) that the condition \((*)\) implies \(f=L(g)\) with a Möbius transformation \(L\). The conclusion of Theorem 4 then follows immediately. The proof in the present paper is almost identical to the proof of G. Brosch.

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D30 Meromorphic functions of one complex variable (general theory)
30D20 Entire functions of one complex variable (general theory)
sharing values
Full Text: DOI
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