Some inequalities for the matrix Heron mean

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$P_t(A, B)$isthe$t$-powermeanof$A$and$B$. As a consequence, we obtain the determinant inequality for the matrix Heron mean: for any positive definite matrices $A$ and $B,$ $$\text{Dt}(A+B+2(A\mathrm{♯}B))\le \text{Dt}(A+B+A^{1\mathrm{/}2}{B}^{1\mathrm{/}2}+A^{1\mathrm{/}2}{B}^{1\mathrm{/}2}))\mathrm{.}$$ These results complement those obtained by Bhatia, Lim and Yamazaki (LAA, {\bf 501} (2016) 112-122).