Some Inequalities of differential polynomials II

In this paper, we consider the value distribution of the differential polynomials $f^2{f}^{(k)}-1$ where $k$ is a positive integer, and obtain some estimates only by the reduced counting function. Our result answers a question in (Some inequalities of differential polynomials, Mathematical Inequalities and Applications, $\mathbf{\text{12}}$(2009), no.1, 99–113) completely.