# On The Operator Hermite--Hadamard Inequality

## TL;DR

This paper explores operator Hermite--Hadamard inequalities for convex functions without relying on operator convexity, presenting various forms and applications to norms and means.

## Contribution

It introduces new forms of operator Hermite--Hadamard inequalities that do not depend on operator convexity, expanding the theoretical framework.

## Key findings

- Multiple forms of the inequality are established.
- Applications to norm inequalities are demonstrated.
- Mean inequalities are derived from the new forms.

## Abstract

The main target of this paper is to discuss operator Hermite--Hadamard inequality for convex functions, without appealing to operator convexity. Several forms of this inequality will be presented and some applications including norm and mean inequalities will be shown too.