Generalized Choi–Davis–Jensen’s Operator Inequalities and Their Applications

Abstract

The original Choi–Davis–Jensen’s inequality, known for its extensive applications in various scientific and engineering fields, has inspired researchers to pursue its generalizations. In this study, we extend the Choi–Davis–Jensen’s inequality by introducing a nonlinear map instead of a normalized linear map and generalize the concept of operator convex functions to include any continuous function defined within a compact region. Notably, operators can be matrices with structural symmetry, enhancing the scope and applicability of our results. The Stone–Weierstrass theorem and the Kantorovich function play crucial roles in the formulation and proof of these generalized Choi–Davis–Jensen’s inequalities. Furthermore, we demonstrate an application of this generalized inequality in the context of statistical physics.

Keywords

• Choi–Davis–Jensen’s inequality
• Stone–Weierstrass theorem
• Kantorovich function
• Loewner ordering