A reverse Hölder inequality is established on the space of Kähler 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 Hölder 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 Hölder bound is shown to only depend on p and the dimension of X. This leads to some intriguing relations to Harnack bounds and the partial \(C^{0}\) -estimate. In another direction, universal effective openness results are established for the complex singularity exponents (log canonical thresholds) of \(\omega \) -plurisubharmonic functions on any Fano variety. Finally, another application to K-unstable Fano varieties is given, involving Archimedean Igusa zeta functions.
