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A reverse H\"older inequality is established on the space of K\"ahler metricsin the first Chern class of a Fano manifold X endowed with Darvas L^{p}-Finslermetrics. The inequality holds under a uniform bound on a twisted Riccipotential and extends to Fano varieties with log terminal singularities. Itsproof leverages a "hidden" log-concavity. An application to destabilizinggeodesic rays is provided, which yields a reverse H\"older inequality for thespeed of the geodesic. In the case of Aubin's continuity path on a K-unstableFano variety, the constant in the corresponding H\"older bound is shown to beuniversal (i.e. it only depends on p and the dimension of X). Some intruigingrelations to Harnack bounds and the partial C^{0}-estimate are pointed out.

Reverse H\"older inequalities on the space of K\"ahler metrics of a Fano variety and log-concavity