On entire functions which share one small function CM with their \(k\)th derivative (Q1781900)

The author proves the following result: Let \(f\) be a non-constant entire function such that its Nevanlinna's characteristic functions satisfy \(\overline {N} (r,\frac{1}{f^{(k)}})=S(r,f)\). If \(f\) and its \(k\)-th derivative \(f^{(k)}\) share a small meromorphic function \(a(\not\equiv 0,\infty)\) CM (counting multiplicity), then \[ f-a=(1-P_{k-1}/a)(f^{(k)}-a), \] where \(P_{k-1}\) is a polynomial of degree at most \(k-1\) such that \(1-P_{k-1}/a=e^\beta\) for an entire function \(\beta\). If \(k=1\) and if \(a\) is constant, this result is due to \textit{R. Brück} [Result. Math. 30, 21--24 (1996; Zbl 0861.30032)].