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Mathematics

Some Quantum Estimates of Hermite-Hadamard Inequalities for Quasi-Convex Functions

Hefeng Zhuang, Wenjun Liu, Jaekeun Park

Published Feb 5, 2019• 44 citations
Publisher

Abstract

In this paper, we develop some quantum estimates of Hermite-Hadamard type inequalities for quasi-convex functions. In some special cases, these quantum estimates reduce to the known results.
DOI
10.3390/math7020152

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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.
One of the beautiful and very simple inequalities for a convex function is the Hermite–Hadamard inequality. The concept of ‐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.
Just as the classical f-divergences parameterized on convex functions generalize the classical relative entropy, the quantum quasi-entropies generalize the quantum relative entropy parameterized on operator convex functions. Quantum quasi-entropies satisfy a version of the monotonicity property and are jointly convex in their arguments. We provide the equality conditions for these inequalities for a family of operator convex functions that includes the von Neumann entropies as a special case. Quantum quasi-entropies are defined with an arbitrarily chosen matrix and since the inequalities are true for any choice of such matrices, we show that these inequalities can be interpreted as operator inequalities.
We consider a quantum quasi-relative entropy [Formula: see text] for an operator [Formula: see text] and an operator convex function [Formula: see text]. We show how to obtain the error bounds for the monotonicity and joint convexity inequalities from the recent results for the [Formula: see text]-divergences (i.e. [Formula: see text]). We also provide an error term for a class of operator inequalities, that generalizes operator strong subadditivity inequality. We apply those results to demonstrate explicit bounds for the logarithmic function, that leads to the quantum relative entropy, and the power function, which gives, in particular, a Wigner–Yanase–Dyson skew information. In particular, we provide the remainder terms for the strong subadditivity inequality, operator strong subadditivity inequality, WYD-type inequalities, and the Cauchy–Schwartz inequality.