One of the beautiful and very simple inequalities for a convex function is the Hermite–Hadamard inequality. The concept of log‐convexity is a strong variant of convexity. In this paper, by the Hermite–Hadamard inequality, we introduce two‐parametric Tsallis quantum relative entropy, two‐parametric Tsallis–Lin quantum relative entropy, and two‐parametric quantum Jensen–Shannon divergence in quantum information theory. Then some properties of quantum Tsallis–Jensen–Shannon divergence for two density matrices are investigated by this inequality.

One of the beautiful and very simple inequalities for a convex function is the Hermit-Hadamard inequality [S. Mehmood, et. al. Math. Methods Appl. Sci., 44 (2021) 3746], [S. Dragomir, et. al., Math. Methods Appl. Sci., in press]. The concept of log-convexity is a stronger property of convexity. Recently, the refined Hermit-Hadamard’s inequalities for log-convex functions were introduced by researchers [C. P. Niculescu, Nonlinear Anal. Theor., 75 (2012) 662]. In this paper, by the Hermit-Hadamard inequality, we introduce two parametric Tsallis quantum relative entropy, two parametric Tsallis-Lin quantum relative entropy and two parametric quantum Jensen-Shannon divergence in quantum information theory. Then some properties of quantum Tsallis-Jensen-Shannon divergence for two density matrices are investigated by this inequality. \newline \textbf{Keywords:} \textit{ Hermit-Hadamard’s inequality; log-convexity; Density matrices; Quantum relative entropy; Tsallis quantum relative entropy; quantum Jensen-Shannon divergence divergence.