Weak Log-Majorization for Negative Lim-P\'alfia Power Means

For $0\le \alpha \le 1$ and $-1\le r<0$, let ${\mathrm{\#}}_{r,\alpha }$ denote the weighted Kubo-Ando power mean of order $r$ and weight $\alpha$. Ando proved that, for every unitarily invariant norm, $\mathrm{\parallel }A{\mathrm{\#}}_{r,\alpha }B\mathrm{\parallel }\le \mathrm{\parallel }A\mathrm{\parallel }{\mathrm{\#}}_{r,\alpha }\mathrm{\parallel }B\mathrm{\parallel }$. 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-P'alfia power means of negative order. It yields analogous Schatten quasi-norm inequalities for every finite family of positive definite matrices.