A new characterization of Gundersen's example of two meromorphic functions sharing four values
Two functions meromorphic and nonrational in the plane are said to share the value \(a \in \widehat\mathbb{C} IM\) (ignoring multiplicities), if \(f^{- 1} (\{a\})=g^{-1}(\{a\})\) holds. If, in addition, every \(p\)-fold \(a\)- point \(z_ 0\) of \(f\) is also a \(p\)-fold \(a\)-point of \(g\), \(p=p(z_ 0)\), then \(IM\) is replaced by \(CM\) (counting multiplities). The four-point- theorem of \(R\). Nevanlinna states that, if \(f\) and \(g\) are distinct meromorphic functions sharing four values \(a_ 1,\dots,a_ 4CM\), then \(f=T\circ g\) with some Möbius transform \(T\), the values \(a_ 3,a_ 4\), say, are Picard exceptional values of both functions and \((a_ 1, a_ 2, a_ 3, a_ 4)=-1\) holds. It is known that the assumptions can be weakened: only two of the values are to be shared \(CM\). However, the theorem is no longer true for functions sharing four values \(IM\). This was shown by \textit{G. G. Gundersen} [J. Lond. Math. Soc. 20, 456-466 (1979; Zbl 0413.30025)]: \[ f(z)={e^{h(z)}+1 \over(e^{h(z)} -1)^ 2} \text{ and } g(z)={(e^{h(z)}+1)^ 2 \over 8 (e^{h(z)} -1)},\;h \not\equiv \text{const} \] entire, share the values \(0,1,\infty\) and \(- 1/8\). In the paper under review it is shown that Gundersen' pair of functions is uniquely determined, up to a Möbius transform, by the following condition: \(f\) and \(g\) share \(a_ 1,\dots,a_ 4\) and there are values \(a,b \neq a_ j\) such that \(f(z)=a\) always implies \(g(z)=b\). In Gundersen's example these values are \(-1/2\) and \(1/2\).
scientific article; zbMATH DE number 446744