  A reverse H\"older inequality is established on the space of K\"ahler metrics
in the first Chern class of a Fano manifold X endowed with Darvas L^{p}-Finsler
metrics. The inequality holds under a uniform bound on a twisted Ricci
potential and extends to Fano varieties with log terminal singularities. Its
proof leverages a "hidden" log-concavity. An application to destabilizing
geodesic rays is provided, which yields a reverse H\"older inequality for the
speed of the geodesic. In the case of Aubin's continuity path on a K-unstable
Fano variety, the constant in the corresponding H\"older bound is shown to be
universal (i.e. it only depends on p and the dimension of X). Some intruiging
relations to Harnack bounds and the partial C^{0}-estimate are pointed out.

Differential Geometry;Algebraic Geometry;Complex Variables