Alpha invariants and coercivity of the Mabuchi functional on Fano manifolds

We give a criterion for the coercivity of the Mabuchi functional for general K"ahler classes on Fano manifolds in terms of Tian’s alpha invariant. This generalises a result of Tian in the anti-canonical case implying the existence of a K"ahler-Einstein metric. We also prove the alpha invariant is a continuous function on the K"ahler cone. As an application, we provide new K"ahler classes on a general degree one del Pezzo surface for which the Mabuchi functional is coercive.