Operator inequalities related to weak 2-positivity

TL;DR

This paper introduces the concept of weak 2-positivity in operators, explores related inequalities, and applies them to derive operator versions of Hua's inequality, expanding the theoretical framework of operator inequalities.

Contribution

The paper defines weak 2-positivity, provides examples, and establishes new operator inequalities involving the geometric mean and Hua's inequality.

Findings

Introduction of weak 2-positivity concept

Operator Cauchy--Schwarz inequalities involving geometric mean

Operator versions of Hua's inequality

Abstract

In this paper we introduce the notion of weak 2-positivity and present some examples. We establish some operator Cauchy--Schwarz inequalities involving the geometric mean and give some applications. In particular, we present some operator versions of Hua's inequality by using the Choi--Davis--Jensen inequality.