For positive definite matrices A and B, the Araki–Lieb–Thirring inequality amounts to an eigenvalue log-submajorisation relation for fractional powersλ(AtBt)≺w(log)λt(AB),0<t⩽1,while for t⩾1, the reversed inequality holds. In this paper I generalise this inequality, replacing the fractional powers xt by a larger class of functions. I prove that a continuous, non-negative function f with domain dom(f)=[0,x0) for some positive x0 (possibly infinity) satisfiesλ(f(A)f(B))≺w(log)f2(λ1/2(AB)),for all positive semidefinite A and B with spectrum in dom(f), if and only if f is geometrically concave and 0⩽xf′(x)⩽f(x) for all x∈dom(f). The reversed inequality holds for a continuous, non-negative function f if and only if f is geometrically convex and xf′(x)⩾f(x) for all x∈dom(f). As an application I obtain a complementary inequality to the Golden–Thompson inequality.