In this paper, we first introduce some properties of quantum Tsallis-Jensen-Shannon divergence for two density matrices. Then one of the beautiful and very simple applications of Hermit-Hadamard’s inequality [S. Dragomir, et. al. Math. Methods Appl. Sci., 1–15 (2020)] for log-convexity in the concepts of quantum entropies and quantum Tsallis-Jensen-Shannon is given.