  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.
