Two entire functions sharing four small functions in the sense of \(\overline E_{k)}(\beta,f)=\overline E_{k)}(\beta,g)\). (Q1871509)

Let \(f\) and \(g\) be two nonconstant entire functions. A Nevanlinna's theorem shows that if there are four distinct complex numbers \(a_1,a_2,a_3\) and \(a_4\) such that \(f^{-1}(a_j)=g^{-1}(a_j)\) for \(j=1,\dots,4\), that is, \(f\) and \(g\) share each \(a_j\) IM (ignoring multiplicity), then \(f=g\). \textit{B. Q. Li} [Am. J. Math. 119, 841--858 (1997)] and \textit{Y. H. Li} [Acta Math. Sin. 41, 249--260 (1998)] proved that the Nevanlinna's theorem holds if replacing four values \(a_j\) by four distinct small meromorphic functions with respect to \(f\) and \(g\). The author further weaken the condition in this paper, that is, she proved that \(f=g\) if there are four distinct small meromorphic functions \(a_1,a_2,a_3\) and \(a_4\) with respect to \(f\) and \(g\) such that the set \(\overline{E}_{k)}(a_j,f)\) of zeros of \(f-a_j\) of order \(\leq k(\geq 11)\) is same with \(\overline{E}_{k)}(a_j,g)\) for \(j=1,\dots,4\).