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 $q_0(\ge 2),q_k(\ge 1)$ are $k(\ge 1)$ non-negative integers. We obtain a quantitative estimation of the characteristic function $T(r,f)$ in terms of $\bar{N}(r,\frac{1}{f^{q_{{}_0}}(f^{(k)}{)}^{q_k}-1})$.\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.