Hlawka-Popoviciu inequalities on positive definite tensors

TL;DR

This paper establishes new inequalities for positive definite tensors and operators, generalizing classical inequalities like Hlawka and Popoviciu, with applications to determinants, permanents, and matrix functions.

Contribution

It introduces multivariable operator inequalities on positive definite tensors, extending known inequalities to a broader tensor and operator context.

Findings

Derived generalized Hlawka and Popoviciu inequalities for determinants and permanents.

Unified several recent inequalities on positive definite matrices as special cases.

Provided new operator inequalities applicable to tensor sums of positive definite operators.

Abstract

We prove inequalities on symmetric tensor sums of positive definite operators. In particular, we prove multivariable operator inequalities inspired by generalizations to the well-known Hlawka and Popoviciu inequalities. As corollaries, we obtain generalized Hlawka and Popoviciu inequalities for determinants, permanents, and generalized matrix functions. The new operator inequalities and their corollaries contain a few recently published inequalities on positive definite matrices as special cases.