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

MSC:

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

Keywords:

Nevanlinna deficiency; d-points
Full Text:

References:

 [1] GROSS, F., On the distribution of values of meromorphic functions, Trans. Amer. Math. Soc, 131(1968), 199-214. · Zbl 0157.12903 [2] GROSS, F. AND C. F. OSGOOD, Entire functions with common preimages, Factorization theory of meromorphic functions, 19-24, Marcel Dekker, Inc., 1982. · Zbl 0494.30029 [3] GROSS, F. AND C. C. YANG, Meromorphic functions covering certain finite sets a the same points, Illinois J. Math., 26 (1982), 432-441. · Zbl 0503.30029 [4] GUNDERSEN, G. G., Meromorphic functions that share three or four values, J. London Math. Soc, (2), ?0 (1979), 457-466 · Zbl 0413.30025 [5] HAYMAN, W. K., Meromorphic Functions (Clarendon Press, Oxford, 1964) · Zbl 0115.06203 [6] HIROMI, G. AND M. OZAWA, On the existence of analytic mappings between two ultrahyperelliptic surfaces, Kodai Math. Sem. Rep., 17(1965), 281-306. · Zbl 0154.07903 [7] MUES, E., Uber die Nullstellen homogener Differentialpolynome, manuscripta math., 23 (1978), 325-341 · Zbl 0382.30021 [8] NEVANLINNA, R., Einige Eindeutigekeitssatze in der Theorie der Meromorphe Funktionen, Acta. Math., 48(1926), 367-391. · JFM 52.0323.03 [9] NEVANLINNA, R., Le theoreme de Picard-Borel et la theorie desfunctions mero morphes (Gauthier-Villars, Paris, 1929). [10] NIINO, K. AND M. OZAWA, Deficiencies of an entire algebroid function, Koda Math. Sem. Rep., 22(1970), 98-113. · Zbl 0199.13002 [11] OZAWA, M., On the zero-one set of an entire function , Kodai Math. J., (1979), 194-199. · Zbl 0416.30026 [12] RUBEL, L. A. AND C. C. YANG, On zero-one sets for entire functions, Mich. Math J., 20 (1973), 289-296. · Zbl 0263.30022 [13] YANG, C. C, On two entire functions which together with their first derivative has the same zeros, J. Math. Anal. Appl., 56 (1976), 1-6. · Zbl 0338.30018
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