On some tensor inequalities based on the t-product

TL;DR

This paper explores tensor inequalities within the t-product framework, extending classical matrix inequalities such as AM-GM, Hölder, and Minkowski to tensors, and establishing t-eigenvalue inequalities.

Contribution

It generalizes key matrix inequalities to tensors in the t-product formalism and proves their validity, advancing tensor analysis theory.

Findings

Tensor power inequalities hold similarly to matrices.

Generalizations of AM-GM, Hölder, and Minkowski inequalities to tensors.

Derivation of t-eigenvalue inequalities.

Abstract

In this work, we investigate the tensor inequalities in the tensor t-product formalism. The inequalities involving tensor power are proved to hold similarly as standard matrix scenarios. We then focus on the tensor norm inequalities. The well-known arithmetic-geometric mean inequality, H{\" o}lder inequality, and Minkowski inequality are generalized to tensors. Furthermore, we obtain some t-eigenvalue inequalities.