Some applications of the Hermit-Hadamard inequality for log-convex
functions in quantum divergence
Abstract
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.