New Improvements to Heron and Heinz Inequality Using Matrix Techniques

Abstract

This paper presents a comprehensive study on matrix means interpolation and comparison, extending the parameter ϑ from the traditional closed interval [0,1] to encompass the entire positive real line, denoted as R+. The research delves into further results involving Heinz means, proposing novel scalar adaptations of Heinz inequalities that integrate Kantorovich’s constant. Additionally, the operator version of these inequalities is strengthened. A key contribution of this work is the development of refined Young’s type inequalities tailored for the traces, determinants, and norms of positive semi-definite matrices. These refinements offer deeper insights into matrix analysis, especially in the context of operator theory and inequality theory. Through these advancements, the paper enhances the mathematical framework for studying matrix means and their associated inequalities, providing useful tools for both theoretical exploration and practical applications in linear algebra and related fields.

Keywords: Matrix means interpolation, Heinz inequalities, Kantorovich’s constant, Operator inequalities, Young’s type inequalities