Some inequalities for the matrix Heron mean

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).