# zbMATH — the first resource for mathematics

Uniqueness of meromorphic functions that share three values. (English) Zbl 0767.30026
The author proves the following theorem on shared values of meromorphic functions. Theorem. Let $$f$$ and $$g$$ be two distinct nonconstant meromorphic functions such that $$f$$ and $$g$$ share 0, 1, $$\infty$$ with the same multiplicities (i.e. C.M.), and let $$a_ 1,a_ 2,\dots,a_ p$$ be $$p$$ ($$\geq 1$$) distinct finite complex numbers, and $$a_ i\neq 0$$ ($$i=1,2,\dots,p$$). If $\sum_{i=1}^ p f(a_ i,f)+f(\infty,f)>{{2(p+1)} \over {p+2}},$ then there exists one and only one $$a_ j$$ in $$a_ 1,a_ 2,\dots,a_ p$$ such that $$a_ j$$ and $$1- a_ j$$ are Picard exceptional values of $$f$$ and $$g$$ respectively, and also $$\infty$$ is so, and $(f-a_ j)(g+a_ j-1)\equiv a_ j(1-a_ j).$ {}.

##### MSC:
 30D30 Meromorphic functions of one complex variable (general theory) 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
##### Keywords:
shared values; Picard exceptional values
Full Text:
##### References:
 [1] G. G. GUNDERSEN, Meromorphic functions that share three or four values, J. London Math. Soc, (2), 20 (1979), 457-466. · Zbl 0413.30025 [2] W. K. HAYMAN, Meromorphic functions, Clarendon Press. Oxford, 1964 · Zbl 0115.06203 [3] H. UEDA, Unicity theorems for meromorphic or entire functions, Kodai Math. J., 3 (1980), 457-471 · Zbl 0468.30023 [4] HONG-XUN Yi, Meromorphic functions that share three values, Chm. Ann. Math., 9A (1988), 434-439 · Zbl 0699.30024 [5] F. GROSS, Factorization of meromorphic functions, U. S. Govt. Printing Offic Publication, Washington D. C., 1972. · Zbl 0266.30006
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.