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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

Citations:

Zbl 0413.30025
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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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