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.


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


Zbl 0413.30025
Full Text: DOI


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