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Meromorphic functions with common preimages. (Chinese. English summary) Zbl 0689.30020
This paper deals with a special case of the general question (raised by F. Gross): Given some finite sets \(S_ i\) \((i=1,2,...,k)\), under what circumstances can two meromorphic functions have identical preimages of \(S_ i\) \((i=1,2,...,k)\). The preimage of a set S for a function f is defined to be \(E_ f(S)=\cup_{a\in S}\{z:\) \(f(z)-a=0\}\) in which a zero of multiplicity m is counted m times. The main result of the paper is that if f, g are two meromorphic functions, and \(S_ 1=\{-1,1\}\), \(S_ 2=\{0\}\), \(S_ 3=\{\infty \}\) such that \(E_ f(S_ i)=E_ g(S_ i)\) \((i=1,2,3)\), then either \(f\equiv \pm g\) or \(fg\equiv \pm 1\). This extends a result obtained earlier by F. Gross and C. F. Osgood [Factorization theory of meromorphic functions and related topics, Lect. Notes Pure Appl. Math. 78, 19-24 (1982; Zbl 0494.30029)]; there f and g are assumed to be two entire functions of finite order. Recently, K. Tohge [Kodai Math. J. 11, No.2, 249-279 (1988; Zbl 0663.30024)] obtained a more general reslt for considering \(S_ 1=\{\xi \in C:\) \(\xi^ n=1\), \(n\geq 2\}\), \(S_ 2=\{0\}\), and \(S_ 3=\{\infty \}\) and some interesting results related to Gross’ question. An essential ingredient in the proof of these types of results is the impossibility of Borel’s identity.

MSC:
30D30 Meromorphic functions of one complex variable, general theory
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