Some Hadamard product inequalities for accretive matrices

TL;DR

This paper establishes new inequalities involving the Hadamard product for accretive and positive definite matrices, extending matrix inequality theory with applications to matrix means and linear maps.

Contribution

It introduces novel Hadamard product inequalities for accretive matrices, including bounds involving matrix means, positive unital linear maps, and matrix concave functions.

Findings

Derived inequalities for positive definite matrices involving Hadamard product

Extended matrix inequality results to accretive matrices

Provided specific bounds for matrix combinations with Hadamard product

Abstract

In this paper, we obtain some new matrix inequalities involving Hadamard product. Also some Hadamard product inequalities for accretive matrices involving the matrix means, positive unital linear maps and matrix concave functions are investigated. Among other results, it is shown that if $A, B, C, D$ are $n\times n$ positive definite matrices, then \begin{equation*} \left(\alpha A+\beta B\right)^r\circ\left(\alpha C+\beta D\right)^{1-r}\leq \alpha\left(A^r\circ C^{1-r}\right)+\beta\left(B^r\circ D^{1-r}\right), \end{equation*} where $r \in (-1, 0) \cup (1, 2)$ and $" \circ "$ stands for the Hadamard product.