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 $Z_{ij}=B_i^{{}^{\frac{1}{{}_2}}}\left(\sum _{k=1}^m{A}_k\right)B_j^{{}^{\frac{1}{{}_2}}}$ for every $i,j=1,\cdots ,m$. It is shown that $$\left|\left|\left|\sum _{i=1}^m{\left(A_i^s♯B_i^s\right)}^r\right|\right|\right|\le \left|\left|\left|Z^{{}^{\frac{sr}{{}_2}}}\right|\right|\right|\le \left|\left|\left|{\left({\left(\sum _{i=1}^m{A}_i\right)}^{\frac{srp}{{}_4}}{\left(\sum _{i=1}^m{B}_i\right)}^{\frac{srp}{{}_2}}{\left(\sum _{i=1}^m{A}_i\right)}^{\frac{srp}{{}_4}}\right)}^{\frac{1}{{}_p}}\right|\right|\right|,$$ for all $s\ge 2$, for all $p>0$ and $r\ge 1$ such that $rp\ge 1$ 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 $s\ge 2$ and for all $p>0$ and $r\ge 1$ such that $rp\ge 1$ 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}.