Uniqueness theorems for meromorphic functions that share four small functions (Q2735465)

\textit{M. Shirosaki} [Tôhoku Math. J. II. Ser. 45, No. 4, 491-497 (1993; Zbl 0802.30026)], \textit{P. Li} and \textit{C. C. Yang} [Complex Variables, Theory Appl. 32, No. 2, 177-190 (1997; Zbl 0880.30029)] extended the four-value theorem of Nevanlinna as follows: Let \(f\) and \(g\) be non-constant meromorphic functions on \(\mathbb{C}\) and let \(a_1\), \(a_2\), \(a_3\), \(a_4\) be four distinct meromorphic functions on \(\mathbb{C}\) such that their Nevanlinna characteristic functions satisfy NEWLINE\[NEWLINET(r,a_j)=o(T(r,f)),\qquad j=1,2,3,4NEWLINE\]NEWLINE except for a possible exceptional set of finite linear measure, that is, \(a_j\) is a small function of \(f\) for \(j=1,\dots ,4\). If \(f\) and \(g\) share \(a_1\), \(a_2\), \(a_3\), \(a_4\) by counting multiplicities, then NEWLINE\[NEWLINEf=\frac{b_1g+b_2}{b_3g+b_4},NEWLINE\]NEWLINE where \(b_j\;(1\leq j\leq 4)\) are small meromorphic functions of \(f\) on \(\mathbb{C}\). Furthermore, for each \(h\in\{f,g\}\), if there exists a number \(u_h\) with \(0\leq u_h<1/19\) such that Nevanlinna functions satisfy NEWLINE\[NEWLINE\bar{N}(r,h)\leq (u_f+o(1))T(r,h)NEWLINE\]NEWLINE except for a possible exceptional set of finite linear measure, \textit{K. Ishizarki} and \textit{N. Toda} [Kodai Math. J. 21, 350-371 (1998; Zbl 0946.30019)] proved \(f=g\). The author shows that the Ishizarki-Toda's result is true when \(0\leq u_h<1/16\) for \(h\in\{f,g\}\).