Operator inequalities of Jensen type

TL;DR

This paper develops generalized Jensen-type inequalities for sequences of self-adjoint operators, providing bounds involving convex functions and operator sums, extending classical inequalities in operator theory.

Contribution

It introduces new operator inequalities of Jensen type involving sequences of self-adjoint operators and convex functions, generalizing existing results.

Findings

Established inequalities for convex functions of operators

Derived bounds involving operator sums and specific functions

Extended Jensen inequalities to operator sequences with bounds

Abstract

We present some generalized Jensen type operator inequalities involving sequences of self-adjoint operators. Among other things, we prove that if $f:[0,\infty) \to \mathbb{R}$ is a continuous convex function with $f(0)\leq 0$, then {equation*} \sum_{i=1}^{n} f(C_i) \leq f(\sum_{i=1}^{n}C_i)-\delta_f\sum_{i=1}^{n}\widetilde{C}_i\leq f(\sum_{i=1}^{n}C_i) {equation*} for all operators $C_i$ such that $0 \leq C_i\leq M \leq \sum_{i=1}^{n} C_i $ \ $(i=1,...,n)$ for some scalar $M\geq0$, where $ \widetilde{C_i} = 1/2 - |\frac{C_i}{M}- 1/2 |$ and $\delta_f = f(0)+f(M) - 2 f(\frac{M}{2})$.