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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