Meromorphic functions covering certain finite sets at the same points. (English) Zbl 0663.30024

This paper provides several interesting results on the general question (raised by 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 of a function f is defined to be \[ E_ f(S)=\cup_{a\in S}\{\xi | \quad f(\xi)-a=0\}, \] in which a zero of multiplicity m is counted m times. Thus \(E_ f(s)=E_ g(s)\) is equivalent to say that f(z)\(\in S\) iff g(z)\(\in S\) CM (counting multiplicities).
In general, it needs five preimage sets of five distinct values to determine a meromorphic function uniquely. However, under certain additional conditions such as multiplicities of the roots of the values or restricted growth rate relationships of two functions can be somewhat determined if they share three values (or three sets each contains no more than two elements) CM. One such result is:
Let n be an integer (n\(\geq 2)\) and \(S_ 1=\{\zeta |\) \(\zeta^ n=1\}\), \(S_ 2=\{0\}\), and \(S_ 3=\{\infty \}\). Suppose that \(f\in S_ i\) iff \(g\in S_ i\) CM for two meromorphic functions f and g. Then either \(f(z)=dg(z)\), where d is a constant with \(d^ n=1\) or \(f^ n(z)=\exp \{\alpha (z)\}\) and \(g^ n(z)=\exp \{-\alpha (z)\}\) for some entire function \(\alpha\). This is a generalization of a result of Xi Hongxun [J. Math. Wuhan Univ. 7, No.3, 217-224 (1987)], where \(n=2\) is treated. Some results of Gross, Yang and Ozawa were also generalized in the paper.
Reviewer: C.-C.Yang


30D30 Meromorphic functions of one complex variable (general theory)
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