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New Improvements to Heron and Heinz Inequality Using Matrix Techniques

M. H. M. Rashid, Wael Mahmoud Mohammad Salameh

TL;DR

The paper addresses interpolation and comparison of matrix means, notably extending the Heron and Heinz frameworks by allowing the interpolation parameter to vary over $\mathbb{R}^+$. It introduces Kantorovich-constant-based scalar Heinz inequalities, extends these to operator settings, and derives refined Young-type inequalities for traces, determinants, and norms of positive semidefinite matrices. Key contributions include full interpolation of matrix variants beyond $[0,1]$, Kantorovich-enhanced Heinz-type refinements, and broad applications to traces, determinants, and norm inequalities. The results broaden the parameter range and sharpen inequalities, with potential impacts in matrix analysis, quantum information, and numerical mathematics.

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.

New Improvements to Heron and Heinz Inequality Using Matrix Techniques

TL;DR

The paper addresses interpolation and comparison of matrix means, notably extending the Heron and Heinz frameworks by allowing the interpolation parameter to vary over . It introduces Kantorovich-constant-based scalar Heinz inequalities, extends these to operator settings, and derives refined Young-type inequalities for traces, determinants, and norms of positive semidefinite matrices. Key contributions include full interpolation of matrix variants beyond , Kantorovich-enhanced Heinz-type refinements, and broad applications to traces, determinants, and norm inequalities. The results broaden the parameter range and sharpen inequalities, with potential impacts in matrix analysis, quantum information, and numerical mathematics.

Abstract

This paper undertakes a thorough investigation of matrix means interpolation and comparison. We expand the parameter beyond the closed interval to cover the entire positive real line, denoted as . 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.
Paper Structure (6 sections, 28 theorems, 85 equations)

This paper contains 6 sections, 28 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Full Interpolation of Matrix Variants of Heron and Heinz MEANS
  3. Sharpening of the Heinz inequalities and its reverses with the Kantorovich Constant
  4. New operator versions of Heinz-type inequalities
  5. Utilizations of the improved Young-type inequalities for traces, determinants, and norms of positive definite matrices
  6. Conclusion and Future Work

Key Result

Theorem 2.1

HJ Let $T,S\in M_n(\mathbb{C})$ such that $T$ is a positive semi-definite. Then where $t_{ii}$ for $i=1,2,\cdots,n$ are the diagonal entries of matrix $T$.

Theorems & Definitions (46)

  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Corollary 2.5
  • Corollary 2.6
  • proof
  • Corollary 2.7
  • Corollary 2.8
  • ...and 36 more