Weak Log-Majorization for Negative Lim-Pálfia Power Means

Abstract

For $0\le \alpha \le 1$ and $-1\le r<0$, let ${\#}_{r,\alpha }$ denote the weighted Kubo-Ando power mean of order $r$ and weight $\alpha$. Ando proved that, for every unitarily invariant norm, $\Vert A{\#}_{r,\alpha }B\Vert \le \Vert A\Vert {\#}_{r,\alpha }\Vert B\Vert$. We show that this inequality remains valid also for the Schatten $q$-quasi-norms with $0<q<1$. More generally, our main result resolves a problem recently posed by Hiai and Lim in the stronger form of weak log-majorization for multivariable Lim-Pálfia power means of negative order. It yields analogous Schatten quasi-norm inequalities for every finite family of positive definite matrices.