  In this paper, we consider the value distribution of the differential
polynomials $f^2f^{(k)}-1$ 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, \textbf{12}(2009), no.1, 99--113) completely.
