Further refinements of the Cauchy-Schwarz inequality for matrices

TL;DR

This paper introduces refined matrix Cauchy-Schwarz inequalities using integration and Hermite-Hadamard techniques, providing tighter bounds for positive semidefinite matrices with potential applications in matrix analysis.

Contribution

It presents new refinements of the matrix Cauchy-Schwarz inequality leveraging integration methods and Hermite-Hadamard inequalities, extending existing bounds for positive semidefinite matrices.

Findings

Established new matrix inequalities involving parameters s, t, and r.

Provided bounds that improve upon classical Cauchy-Schwarz inequalities.

Extended the inequality framework to include integration-based refinements.

Abstract

Let $A, B$ and $X$ be $n\times n$ matrices such that $A, B$ are positive semidefinite. We present some refinements of the matrix Cauchy-Schwarz inequality by using some integration techniques and various refinements of the Hermite--Hadamard inequality. In particular, we establish the inequality \begin{align*} |||\,|A^{1\over2}XB^{1\over2}|^r|||^2&\leq|||\,|A^{t}XB^{1-s}|^r||| \,\,\,|||\,|A^{1-t}XB^{s}|^r|||\\& \leq\max \{|||\,|AX|^r||| \,\,\,|||\,|XB|^r|||,|||\,|AXB|^r||| \,\,\,|||\,|X|^r|||\}, \end{align*} where $s,t\in[0,1]$ and $r\geq0$.