# Operator Jensen's inequality for operator superquadratic functions

## TL;DR

This paper introduces the concept of operator superquadratic functions for positive Hilbert space operators, explores their properties, and establishes a non-commutative Jensen's inequality with generalizations.

## Contribution

It defines operator superquadratic functions, provides examples, and proves a non-commutative Jensen's inequality with extensions to positive unital linear maps.

## Key findings

- Defined operator superquadratic functions with key properties
- Established a non-commutative Jensen's inequality for these functions
- Generalized the inequality to positive unital linear maps

## Abstract

In this work, an operator superquadratic function (in operator sense) for positive Hilbert space operators is defined. Several examples with some important properties together with some observations which are related to the operator convexity are pointed out. Equivalent statements of a non-commutative version of Jensen's inequality for operator superquadratic function are established. A generalization of the main result to any positive unital linear map is also provided.