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

A strengthened monotonicity inequality of quantum relative entropy: A unifying approach via R\'{e}nyi relative entropy