# On the Operator Jensen Inequality for Convex Functions

## TL;DR

This paper extends the operator Jensen inequality to a broader class of convex functions, including non-operator convex functions, providing new generalizations and their reverses, with various special cases analyzed.

## Contribution

It introduces a novel generalization of the Jensen inequality for convex functions beyond operator convexity, including reverse inequalities.

## Key findings

- Established a new operator Jensen inequality for convex functions.
- Derived reverse inequalities related to the new Jensen inequality.
- Discussed several special cases of the generalized inequalities.

## Abstract

This paper is mainly devoted to studying operator Jensen inequality. More precisely, a new generalization of Jensen inequality and its reverse version for convex (not necessary operator convex) functions have been proved. Several special cases are discussed as well.