The uniqueness of meromorphic functions concerning small functions
In this paper the authors extend the so-called ``five-point-theorem'' into the case of five ``small'' functions. This uniqueness theorem due to R. Nevanlinna states that any non-constant function meromorphic in the plane can be uniquely determined by its preimages of arbitrary five distinct points. Nevanlinna's proof of this result is one of the most interesting applications of his own second fundamental theorem on value distribution. It is independently and almost at the same time that \textit{C. F. Osgood} [J. Number Theory 21, 347-389 (1985; Zbl 0575.10032)] and \textit{N. Steinmetz} [J. Reine Angew. Math. 368, 134-141 (1986; Zbl 0598.30045)] have shown their generalized second fundamental theorem on ``moving targets''. In these papers or the paper of \textit{G. Frank} and \textit{G. Weissenborn} [Bull. Lond. Math. Soc. 18, 29-33 (1986; Zbl 0586.30025)], for example, we learn that some functional determinants, especially Wronskians, have importance as counterparts of ordinary derivatives in the classical case. But for the solution given in the present paper, it is not enough to apply this generalized second fundamental theorem due to Osgood and Steinmetz, since any ramification term is not included in their estimate as in Nevanlinna's [see \textit{G. Frank} and \textit{G. Weissenborn} [Complex Variables, Theory Appl. 12, No. 1-4, 77-81 (1989; Zbl 0683.30031) for such an estimate]. The authors prove the generalized uniqueness theorem by applying a ``cut-off'' version (not the above-mentioned ``truncated'' one due to Frank and Weissenborn) of the Osgood-Steinmetz second fundamental theorem (Lemma~3) among others. This result is originally given by \textit{Q. Zhang} [Acta Math. Sin. 36, No. 6, 826-833 (1993; Zbl 0798.30020)]. They introduce several well-constructed auxiliary functional determinants as well as Zhang, the first author [Acta Math. Sin. 41, No.2, 249-260 (1998)] and others. The discussion proceeds by repeatedly taking linear fractional transformations of functions under consideration and also a nest of subsets~\(I\) of the positive real axis with infinite linear measure. (A reader might easily lose her/his ``\(I\)'' in a series of the \(I\)'s. For example, the~\(I\) in the expression (3.9) should be read as \(I_q\), a subset of the~\(I\) in (3.8), with infinite linear measure depending on the integer~\(q\).).
scientific article; zbMATH DE number 1531265