Matrix inequalities for the difference between arithmetic mean and harmonic mean

TL;DR

This paper generalizes inequalities comparing the arithmetic and harmonic means for scalars and matrices, including new bounds involving the Hilbert-Schmidt norm and determinant.

Contribution

It introduces novel scalar and matrix inequalities for the difference between arithmetic and harmonic means, extending existing results with new bounds and norm-based inequalities.

Findings

New inequalities for scalar and matrix means

Bounds involving Hilbert-Schmidt norm and determinant

Generalizations of existing mean inequalities

Abstract

Motivated by the refinements and reverses of arithmetic-geometric mean and arithmetic-harmonic mean inequalities for scalars and matrices, in this article, we generalize the scalar and matrix inequalities for the difference between arithmetic mean and harmonic mean. In addition, relevant inequalities for the Hilbert-Schmidt norm and determinant are established.