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 $$\lambda (A^t{B}^t){\prec }_{w(\log )}{\lambda }^t(AB),0<t\le 1,$$ while for $t\ge 1$, 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 $\text{dom}(f)=[0,x_0)$ for some positive $x_0$ (possibly infinity) satisfies $$\lambda (f(A)f(B)){\prec }_{w(\log )}{f}^2({\lambda }^{1\mathrm{/}2}(AB)),$$ for all positive semidefinite $A$ and $B$ with spectrum in $\text{dom}(f)$, if and only if $0\le x{f}^{\mathrm{\prime }}(x)\le f(x)$ for all $x\in \text{dom}(f)$. The reversed inequality holds for continuous, non-negative, geometrically convex functions if and only if they satisfy $x{f}^{\mathrm{\prime }}(x)\ge f(x)$ for all $x\in \text{dom}(f)$. As an application I derive a complementary inequality to the Golden-Thompson inequality.