  We derive a strengthened monotonicity inequality for quantum relative entropy
by employing properties of $\alpha$-R\'{e}nyi relative entropy. We develop a
unifying treatment towards the improvement of some quantum entropy
inequalities. In particular, an emphasis is put on a lower bound of quantum
conditional mutual information (QCMI) as it gives a Pinsker-like lower bound
for the QCMI. We also give some improved entropy inequalities based on
R\'{e}nyi relative entropy. The inequalities obtained, thus, extends some
well-known ones. We also obtain a condition under which a tripartite operator
becomes a Markov state. As a by-product we provide some trace inequalities of
operators, which are of independent interest.
