Meromorphic functions that share three sets. (English) Zbl 0882.30019

Let \(n\) be a positive integer, and let \(w= \exp (2\pi i/n)\). Let \(S_1= \{1,w,w^2, \dots, w^{n-1}\}\), \(S_2 =\{0\}\), and \(S_3= \{\infty\}\). The author proves six theorems which are improvements or supplements to known results. A typical theorem assumes \(f\) and \(g\) are nonconstant meromorphic functions in the plane which share sets \(S_1\) and \(S_3\), counting multiplicity, and \(S_2\) ignoring multiplicity and proves that if \(n\geq 2\), then \(f=tg\) where \(t^n=1\) or \(f\cdot g=s\) where \(s^n=1\).
Reviewer: L.R.Sons (DeKalb)


30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D30 Meromorphic functions of one complex variable (general theory)
30D20 Entire functions of one complex variable (general theory)
Full Text: DOI


[1] W. K. HAYMAN, Meromorphic Functions, Clarendon Press, Oxford, 1964. · Zbl 0115.06203
[2] F. GROSS, On the distribution of values of meromorphic functions, Trans. Amer. Math. Soc, 131 (1968), 199-214 · Zbl 0157.12903
[3] G. G. GUNDENSEN, Meromorphic functions that share three or four values, J. London Math Soc, 20 (1979), 457-466. · Zbl 0413.30025
[4] F. GROSS AND C. F. OSGOOD, Entire functions with common preimages, Factorization Theor of Meromorphic Functions, Marcel Dekker, 1982, 19-24. · Zbl 0494.30029
[5] H. -X. Yi, Meromorphic functions with common preimages, J. of Math. (Wuhan), 7 (1987), 219-224 · Zbl 0689.30020
[6] G. BROSCH, Eindeutigkeitssatze fur meromorphe Funktionen, Thesis, Technical University o Aachen, 1989. · Zbl 0694.30027
[7] H. -X. Yi, On the uniqueness of meromorphic functions, Acta Math. Sinica, 31 (1988), 570-576 · Zbl 0679.30024
[8] K. TOHGE, Meromorphic functions covering certain finite sets at the same points, Kodai Math J., 11 (1988), 249-279. · Zbl 0663.30024
[9] H. -X. Yi, Meromorphic functions that share two or three values, Kodai Math. J., 13 (1990), 363-372 · Zbl 0712.30029
[10] G. JANK AND N. TERGLANE, Meromorphic functions sharing three values, Math. Pannon., (1991), 37-46. · Zbl 0747.30022
[11] L. R. SONS, Deficiencies of monomials, Math. Z., III (1969), 53-68 · Zbl 0176.37204
[12] S. -P. WANG, On meromorphic functions that share four values, J. Math. Anal. Appl., 17 (1993), 359-369. · Zbl 0781.30026
[13] H. -X. Yi AND C. C. YANG, Theory of the Uniqueness of Meromorphic Functions, Scienc Press, Beijing, 1995.
[14] R. NEVANLINNA, Einige Eindeutigkeitssatze in der Theoe der meromorphen Funktionen, Acta Math., 48 (1926), 367-391 · JFM 52.0323.03
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.