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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.

30D30 Meromorphic functions of one complex variable, general theory