Meromorphic functions that share four small functions. (Q2749013)

The author proves the following main result: Let \(f\) and \(g\) be nonconstant meromorphic functions sharing four distinct small functions \(a_1,a_2,a_3,a_4\) ignoring multiplicities. If there is a small function \(a_5\) distinct from \(a_j\), \(j=1,2,3,4\), such that NEWLINE\[NEWLINE\overline{N}(r,f=a_5=g)\not=o(T(r,f))\;(r\to\infty)NEWLINE\]NEWLINE possibly outside a set of \(r\) of finite linear measure, then \(f=g\), where \(T(r,f)\) is the Nevanlinna characteristic function of \(f\), and \(\overline{N}(r,f=a_5=g)\) is the valence function of common zeros of \(f-a_5\) and \(g-a_5\) counted only once (ignoring multiplicities). This result also was obtained by \textit{K. Ishizaki} [Arch. Math. 77, No. 3, 273--277 (2001; Zbl 1093.30027)]. Particularly, if \(f\) and \(g\) share \(a_5\) ignoring multiplicities, the result was proved by \textit{Y. H. Li} and \textit{J. Y. Qiao} [Adv. Math. 28, No. 1, 87--88 (Chinese) (1999), Sci. China, Ser. A 43, No. 6, 581--590 (2000; Zbl 0970.30018)].