Value distribution of some differential monomials

Let $f$ be a transcendental meromorphic function defined in the complex plane $mathbb{C}$. We consider the value distribution of the differential polynomial $f^{q_{0}}(f^{(k)})^{q_{k}}$, where $q_{0}(geq 2), q_{k}(geq 1)$ are $k(geq1)$ non-negative integers. We obtain a quantitative estimation of the characteristic function $T(r, f)$ in terms of $overline{N}left(r,frac{1}{f^{q_{_{0}}}(f^{(k)})^{q_{k}}-1}right)$.par Our result generalizes the results obtained by Xu et al. (Math. Inequal. Appl., 14, 93-100, 2011) and Karmakar and Sahoo (Results Math., 73, 2018) for a particular class of transcendental meromorphic functions.