Some inequalities for the matrix Heron mean

TL;DR

This paper establishes new inequalities involving the matrix Heron mean and related matrix means, providing bounds for determinants and norms of positive definite matrices, extending prior work in matrix analysis.

Contribution

It introduces novel inequalities for the matrix Heron mean and compares different matrix means, extending existing results in matrix inequalities literature.

Findings

Established inequality for matrix Heron mean involving Schatten p-norms.

Proved determinant inequalities for matrix power and Heron means.

Extended previous inequalities by Bhatia, Lim, and Yamazaki.

Abstract

Let $A, B$ be positive definite matrices, $p=1, 2$ and $r\ge 0$. It is shown that \begin{equation*} ||A+ B + r(A\sharp_t B+A\sharp_{1-t} B)||_p \le ||A+ B + r(A^{t}B^{1-t} + A^{1-t}B^t)||_p. \end{equation*} We also prove that for positive definite matrices $A$ and $B$ \begin{equation*}\label{det} \Dt (P_{t}(A, B)) \le \Dt (Q_{t}(A, B)), \end{equation*} where $Q_t(A, B)= \big(\frac{A^t+B^t}{2}\big)^{1/t}$ and $P_t(A, B)$ is the $t$-power mean of $A$ and $B$. As a consequence, we obtain the determinant inequality for the matrix Heron mean: for any positive definite matrices $A$ and $B,$ $$ \Dt(A+ B + 2(A\sharp B)) \le \Dt(A+ B + A^{1/2}B^{1/2} + A^{1/2}B^{1/2})). $$ These results complement those obtained by Bhatia, Lim and Yamazaki (LAA, {\bf 501} (2016) 112-122).