  We provide an upper bound on the quasi-relative entropy in terms of the trace
distance. The bound is derived for two cases: 1) any operator monotone
decreasing function and full rank mixed qubit or classical states; 2) a large
class of operator monotone decreasing function and any mixed qubit or classical
states. Moreover, we derive an upper bound for the Umegaki and Tsallis relative
entropies in the case of any finite-dimensional states. The bound for the
relative entropy improves the known bounds for some states in any dimensions
larger than four. The bound for the Tsallis entropy improves the known bounds.
