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).
Categories: math.FA math.OA