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Generalized Choi-Davis-Jensen's Operator Inequalities and Their Applications

Shih Yu Chang, Yimin Wei

Abstract

The original Choi-Davis-Jensen's inequality, with its wide-ranging applications in diverse scientific and engineering fields, has motivated researchers to explore generalizations. In this study, we extend Davis-Choi-Jensen's inequality by considering a nonlinear map instead of a normalized linear map and generalize operator convex function to any continuous function defined in a compact region. The Stone-Weierstrass theorem and Kantorovich function are instrumental in formulating and proving generalized Choi-Davis-Jensen's inequalities. Additionally, we present an application of this generalized inequality in the context of statistical physics.

Generalized Choi-Davis-Jensen's Operator Inequalities and Their Applications

Abstract

The original Choi-Davis-Jensen's inequality, with its wide-ranging applications in diverse scientific and engineering fields, has motivated researchers to explore generalizations. In this study, we extend Davis-Choi-Jensen's inequality by considering a nonlinear map instead of a normalized linear map and generalize operator convex function to any continuous function defined in a compact region. The Stone-Weierstrass theorem and Kantorovich function are instrumental in formulating and proving generalized Choi-Davis-Jensen's inequalities. Additionally, we present an application of this generalized inequality in the context of statistical physics.
Paper Structure (5 sections, 10 theorems, 70 equations)

This paper contains 5 sections, 10 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Generalized Choi-Davis-Jensen's Inequalities with Error Control
  3. Applications
  4. Tsallis Relative Entropy Without $\Phi$
  5. Tsallis Relative Entropy With $\Phi$

Key Result

Lemma 1

Given a self-adjoint operator $\bm{A}$ with $\Lambda(\bm{A})$, such that for $x \in [\min(\Lambda(\bm{A})), \max(\Lambda(\bm{A}))]$ with polynomials $p_{\mathscr{L}}(x)$ and $p_{\mathscr{U}}(x)$ expressed by Further assume that $p_{\mathscr{L}}(\bm{A})\geq \bm{0}$, we have and where Kantorovich functions $\mathscr{K}^{-1}(\min(\Lambda(p_{\mathscr{L}}(\bm{A}))),\max(\Lambda(p_{\mathscr{L}}(\bm{

Theorems & Definitions (16)

  • Definition 1
  • Lemma 1
  • Remark 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Corollary 1
  • Example 1
  • Example 2
  • Lemma 4
  • ...and 6 more