Norm inequalities involving geometric means

Abstract

Let $A_i$ and $B_i$ be positive definite matrices for every $i=1,\cdots,m.$ Let $Z=[Z_{ij}]$ be the block matrix, where $Z_{ij}=B_i^{^\frac{1}{_2}}\left(\displaystyle\sum_{k=1}^mA_k\right)B_j^{^\frac{1}{_2}}$ for every $ i,j=~1,\cdots,m$. It is shown that $$\left|\left|\left|\sum_{i=1}^m\left(A_i^{s}\sharp B_i^{s}\right)^r\right|\right|\right|\leq\left|\left|\left| Z^{^\frac{sr}{_2}} \right|\right|\right| \leq \left|\left|\left|\left(\left(\sum_{i=1}^mA_i\right)^\frac{srp}{_4}\left(\sum_{i=1}^mB_i\right)^\frac{srp}{_2}\left(\sum_{i=1}^mA_i\right)^\frac{srp}{_4}\right)^{\frac{1}{_p}}\right|\right|\right|,$$ for all $s\geq2$, for all $p>0$ and $r\geq1$ such that $rp\geq1$ and for all unitarily invariant norms. This result generalizes the results in \cite{ONIR} and gives an affirmative answer to a conjecture in \cite{OACRT} for all $s\geq2$ and for all $p>0$ and $r\geq1$ such that $rp\geq1$ and $t=\frac{1}{2}$. This result also leads directly to Dinh, Ahsani, and Tam's conjecture in \cite{GAI} and proves Audenaert's result in \cite{ANIFP}.