New Improvements to Heron and Heinz Inequality Using Matrix Techniques

TL;DR

This paper advances matrix inequalities by extending parameter ranges, introducing new scalar and operator inequalities involving Heinz means, Kantorovich's constant, and refined Young's inequalities for matrices.

Contribution

It extends the parameter range for matrix mean inequalities, introduces new scalar Heinz inequalities with Kantorovich's constant, and refines Young's inequalities for matrix traces, determinants, and norms.

Findings

Extended parameter range for matrix means to all positive reals

Developed new scalar Heinz inequalities involving Kantorovich's constant

Refined Young's inequalities for positive semi-definite matrices

Abstract

This paper undertakes a thorough investigation of matrix means interpolation and comparison. We expand the parameter $\vartheta$ beyond the closed interval $[0,1]$ to cover the entire positive real line, denoted as $\mathbb{R}^+$. Furthermore, we explore additional outcomes related to Heinz means. We introduce new scalar adaptations of Heinz inequalities, incorporating Kantorovich's constant, and enhance the operator version. Finally, we unveil refined Young's type inequalities designed specifically for traces, determinants, and norms of positive semi-definite matrices.