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arXiv · 2608.08850

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

Marco Tomamichel
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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, $\| A \ \#_{r,\alpha}\ B \| \le \| A \|\ \#_{r,\alpha}\ \| B \|$. 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.

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Marco Tomamichel. 2026-08-09. Weak Log-Majorization for Negative Lim-P\'alfia Power Means. https://arxiv.org/abs/2608.08850

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