## On meromorphic functions sharing five one-point or two-point sets IM.(English)Zbl 1205.30032

In this paper, a meromorphic function means a function meromorphic in the whole complex plane $$\mathbb{C}$$. The author studies such possible non-constant meromorphic functions that attain given one-point or two-point sets in the extended complex plane $$\bar{\mathbb{C}}:=\mathbb{C}\cup\{\infty\}$$ at the same points. As the main theorem of this paper, Theorem 1, the author shows that one of the two functions must be a Möbius transformation of the other if certain mutually distinct five one-point or two-point sets in $$\bar{\mathbb{C}}$$ are shared by such functions IM (ignoring multiplicities). Note that two meromorphic functions, say $$f$$ and $$g$$, are said to share a finite set $$S \subset \bar{\mathbb{C}}$$ IM when the preimages of $$S$$ by $$f$$ and by $$g$$ coincide in $$\mathbb{C}$$. The term IM is replaced by CM (counting multiplicities) if, in addition, the following holds: if $$a, b \in S$$ and $$z_0$$ is an $$a$$-point of $$f$$ of multiplicity $$p$$ and a $$b$$-point of $$g$$ of multiplicity $$q$$, then $$p=q$$. Therefore, all the five sets in Theorem 1 are shared by the pair of functions not only IM but also CM, naturally.
The celebrated five-point theorem of R. Nevanlinna states that if two non-constant meromorphic functions share distinct five one-point sets IM, then they must be identical. On the other hand, his four-point theorem states that if $$f$$ and $$g$$ are meromorphic functions sharing distinct four one-point sets CM, then $$f$$ is a Möbius transformation of $$g$$. This is no longer true for functions sharing four values IM only, as G. G. Gundersen [J. Lond. Math. Soc. 20, 456–466 (1979; Zbl 0413.30025)] showed by an interesting counterexample, that is, a pair of meromorphic functions sharing certain four distinct values IM (and never CM in each) which cannot be related with a Möbius transformation. Furthermore, it was proved that certain two other values are attained by Gundersen’s pair at the same points in the following sense: Two meromorphic functions $$f$$ and $$g$$ are said to share a pair of values $$(a, b)$$ in $$\bar{\mathbb{C}}$$, when $$f(z_0) = a$$ if and only if $$g(z_0) = b$$. Here, the choice $$a=b$$ is allowed as special case, that is, as a shared one-point set or a shared value. In addition, one says that $$f$$ and $$g$$ share the (ordered) pair $$(a, b)$$ CM when the following holds: if $$z_0$$ is an $$a$$-point of $$f$$ of multiplicity $$p$$ and a $$b$$-point of $$g$$ of multiplicity $$q$$, then $$p = q$$. Gundersen’s pair shares mutually distinct five pairs of values $$(a_j,b_j)$$ in $$\bar{\mathbb{C}}$$, but, of course, it does not share five one-point or two-point sets $$\{a_j, b_j\}$$ in the sense of the paper under review.
The author applies Nevanlinna’s theory to prove Theorem 1 and gives an interesting conjecture for functions sharing mutually distinct four one-point or two-point sets CM.

### MSC:

 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

### Keywords:

uniqueness theorem; sharing sets; Nevanlinna theory

Zbl 0413.30025
Full Text:

### References:

 [1] W. K. Hayman, Meromorphic functions , Clarendon Press, Oxford, 1964. · Zbl 0115.06203 [2] R. Nevanlinna, Einige Eindeutigkeitssätze in der Theorie der Meromorphen Funktionen, Acta Math. 48 (1926), no. 3-4, 367-391. · JFM 52.0323.03 [3] R. Nevanlinna, Le théorèm de Picard-Borel et la théorie des fonctions méromorphes, Gauthier-Villars, Paris, 1929. · JFM 55.0773.03 [4] M. Shirosaki, A new characterization of collections of two-point sets with the uniqueness property, Kodai Math. J. 30 (2007), no. 2, 213-222. · Zbl 1134.30329 [5] M. Shirosaki and M. Taketani, On meromorphic functions sharing two one-point sets and two two-point sets, Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), no. 3, 32-35. · Zbl 1134.30327 [6] K. Tohge, Meromorphic functions covering certain finite sets at the same points, Kodai Math. J. 11 (1988), no. 2, 249-279. · Zbl 0663.30024
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