An Araki-Lieb-Thirring inequality for geometrically concave and geometrically convex functions

TL;DR

This paper extends the Araki-Lieb-Thirring inequality to a broader class of functions, specifically geometrically concave and convex functions, providing new eigenvalue inequalities for positive definite matrices.

Contribution

It generalizes the classical inequality by replacing fractional powers with geometrically concave or convex functions, characterizing when the inequalities hold based on derivative conditions.

Findings

Established inequalities for geometrically concave functions

Derived reversed inequalities for geometrically convex functions

Provided a new inequality related to the Golden-Thompson inequality

Abstract

For positive definite matrices $A$ and $B$, the Araki-Lieb-Thirring inequality amounts to an eigenvalue log-submajorisation relation for fractional powers $$\lambda(A^t B^t) \prec_{w(\log)} \lambda^t(AB), \quad 0<t\le 1,$$ while for $t\ge1$, the reversed inequality holds. In this paper I generalise this inequality, replacing the fractional powers $x^t$ by a larger class of functions. Namely, a continuous, non-negative, geometrically concave function $f$ with domain $\dom(f)=[0,x_0)$ for some positive $x_0$ (possibly infinity) satisfies $$\lambda(f(A) f(B)) \prec_{w(\log)} f^2(\lambda^{1/2}(AB)),$$ for all positive semidefinite $A$ and $B$ with spectrum in $\dom(f)$, if and only if $0\le xf'(x)\le f(x)$ for all $x\in\dom(f)$. The reversed inequality holds for continuous, non-negative, geometrically convex functions if and only if they satisfy $xf'(x)\ge f(x)$ for all $x\in\dom(f)$. As an application I derive a complementary inequality to the Golden-Thompson inequality.