Entire functions that share one small function (Q2721691)

The authors prove a unicity theorem corresponding to a result of \textit{Q. D. Zhang} [Acta Math Sinica 37, No. 1, 91-98 (1994; Zbl 0792.30022)]. Let \(n\geq 6\) be a positive integer, and let \(f\) and \(g\) be two transcendental entire functions. Let \(a\) be a meromorphic function in the plane with \(a(z)\not\equiv 0\) such that \(T(r,a)= o(T(r,f))\) and \(T(r,a)= o(T(r,g)\) as \(r\to+\infty\), possibly outside a set of finite measure. If \(f^n(z)f'(z)-a(z)\) and \(g^n(z)g'(z) -a(z)\) assume the same zeros with the same multiplicities, then either \(f^nf'g^ng'\equiv a^2\) or \(f\equiv cg\) for a constant \(c\) with \(c^{n+1} =1\). The method of proof is different from that used by the authors in proving the unicity theorem which appears in [J. Nanjing Univ, Math. Biq. 13, No. 1, 44-48 (1996; Zbl 0899.30022)].