##
**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.

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.

Reviewer: Kazuya Tohge (Kanazawa)

### MSC:

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

### Citations:

Zbl 0413.30025
PDFBibTeX
XMLCite

\textit{M. Shirosaki}, Proc. Japan Acad., Ser. A 86, No. 1, 6--9 (2010; Zbl 1205.30032)

Full Text:
DOI

### 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 |

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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.