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

For positive definite matrices $A$ and $B$, the Araki-Lieb-Thirring inequality amounts to an eigenvalue log-submajorisation relation for fractional powers $$λ(A^t B^t) \prec_{w(\log)} λ^t(AB), \quad 0<t\le 1,$$ 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 $$λ(f(A) f(B)) \prec_{w(\log)} f^2(λ^{1/2}(AB)),$$ 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.