A log-majorization version of Audenaert's inequality

Abstract

For $i=1,\dots ,k$, let $A_i$ and $B_i$ be positive definite matrices. It is shown that$s{\left(\sum _{i=1}^m{A}_i{♯}_t{B}_i\right)}^r{\prec }_{\log }s{\left({\left(\sum _{i=1}^m{B}_i\right)}^{\frac{tpr}{2}}{\left(\sum _{i=1}^m{A}_i\right)}^{(1-t)pr}{\left(\sum _{i=1}^m{B}_i\right)}^{\frac{tpr}{2}}\right)}^{\frac{1}{p}}$for all $r>0$, $p>0$ and $t\in [0,1]$. This is stronger than the inequalitywhere $A_i$ commutes with $B_i$ for each i and for all unitarily invariant norms, which has been proved by Audenaert. Applications of these inequalities shed some light on the solution of a question of Bourin.