On the Value Distribution of $f^n{f}^{\left(k\right)}-1$

In this paper, we consider the value distribution of the differential polynomial $f^n{f}^{\left(k\right)}-1$ where $n(\ge 2)$ and $k(\ge 1)$ are integers and obtain some inequalities concerning the Nevanlinna characteristic function T(r, f). Our result improves and generalizes the results obtained by Xu et al. (Math Inequal Appl 14:93–100, 2011).