On meromorphic functions sharing five one-point or two-point sets IM
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 \textit{five} one-point or two-point sets in~\(\bar{\mathbb{C}}\) are shared by such functions \textbf{IM} (ignoring multiplicities). Note that two meromorphic functions, say \(f\) and \(g\), are said to share a finite set \(S \subset \bar{\mathbb{C}}\) \textbf{IM} when the preimages of~\(S\) by~\(f\) and by~\(g\) coincide in~\(\mathbb{C}\). The term \textbf{IM} is replaced by \textbf{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 \textbf{IM} but also \textbf{CM}, naturally. The celebrated \textit{five-point theorem} of R. Nevanlinna states that if two non-constant meromorphic functions share distinct \textit{five} one-point sets \textbf{IM}, then they must be identical. On the other hand, his \textit{four-point theorem} states that if \(f\) and \(g\) are meromorphic functions sharing distinct \textit{four} one-point sets \textbf{CM}, then \(f\) is a Möbius transformation of \(g\). This is no longer true for functions sharing \textit{four} values \textbf{IM} only, as \textit{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 \textit{four} distinct values \textbf{IM} (and never \textbf{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 \textit{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 \((a, b)\) \textbf{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 \textit{five} pairs of values \((a_j,b_j)\) in~\(\bar{\mathbb{C}}\), but, of course, it does not share \textit{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 \textit{four} one-point or two-point sets \textbf{CM}.
scientific article; zbMATH DE number 5690866