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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 $B$. 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