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

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For positive definite matrices A and B, the Araki-Lieb-Thirring inequality amounts to an eigenvalue log-submajorisation relation for fractional powers λ(Aᵗ Bᵗ) () 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. Namely, a continuous, non-negative, geometrically concave function f with domain (f)=[0,x₀) for some positive x₀ (possibly infinity) satisfies λ(f(A) f(B)) () f²(λ¹/²(AB)), for all positive semidefinite A and B with spectrum in (f), if and only if 0≤ xf'(x)≤ f(x) for all x∈(f). The reversed inequality holds for continuous, non-negative, geometrically convex functions if and only if they satisfy xf'(x)≥ f(x) for all x∈(f). As an application I derive a complementary inequality to the Golden-Thompson inequality.