  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.
