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Meromorphic functions that share two or three values. (English) Zbl 0712.30029

The author proves the following: Theorem: Let f and g be meromorphic functions such that f and g share 1,\(\infty\) with the same multiplicity. If \[ N(\gamma,\frac{1}{f})+N(\gamma,\frac{1}{g})+2\bar N(\gamma,f)<(u+o(1))T(\gamma)\quad (\gamma \in I), \] where \(u<1\), then \(f=g\) or \(fg=1\). This result is an extension of earlier results of the reviewer, Osgood, Gunderson and others.
Reviewer: Fred Gross

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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[1] F. GROSS AND C. F. OSGOOD, Entire functions with common preimages, Factori-zation Theory of Meromorphic Functions, 19-24, Marcel Dekker, Inc., 1982. · Zbl 0494.30029
[2] G. G. GUNDERSEN, Meromorphic functions that share three or four values, J. London Math. Soc, (2), 20 (1979), 457-466 · Zbl 0413.30025
[3] W. K. HAYMAN, Meromorphic Functions, Clarendon Press, Oxford, 1964 · Zbl 0115.06203
[4] R. NEVANLINNA, Le Theorems de Picard-Borel et la Theorie des Functions Mero morphes, Gauthier-Villars, Paris, 1929.
[5] M. OZAWA, Unicity theorems for entire functions, J. d’Anal. Math., 30(1976), 411-420 · Zbl 0337.30020
[6] K. TOHGE, Meromorphic functions covering certain finite sets at the same points, Kodai Math. J., 11 (1988), 249-279 · Zbl 0663.30024
[7] H. UEDA, Unicity theorems for entire functions, Kodai Math. J., 3 (1980), 212-223 · Zbl 0442.30024
[8] H. UEDA, Unicity theorems for meromorphic or entire functions, Kodai Math J., 3 (1980), 457-471. · Zbl 0468.30023
[9] H. UEDA, Unicity theorems for meromorphic or entire functions II, Kodai Math J., 6 (1983), 26-36. · Zbl 0518.30029
[10] HONG-XUN Yi, Meromorphic functions with common premages, J. of Math. (PRC), 7 (1987), 219-224 · Zbl 0689.30020
[11] HONG-XUN Yi, Meromorphic functions with two deficient values, Acta Math Sin., 30 (1987), 588-597. · Zbl 0654.30023
[12] HONG-XUN YI, On the uniqueness of meromorphic functions, Acta Math. Sin., 3 (1988), 570-576. · Zbl 0679.30024
[13] HONG-XUN YI, Meromorphic functions that share three values, Chin. Ann. Math., 9A (1988), 434-440 · Zbl 0699.30024
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