  We develop some pluripotential theoretic techniques for the transversally
holomorphic foliation of a Sasakian manifold. We prove the convexity of the
K-energy along weak geodesics for Sasakian manifolds. This implies that the
K-energy is bounded below if a constant scalar curvature structure exists with
those metrics minimizing it. More generally, a relative version of the K-energy
is convex, and bounded below if there exists a Sasaki-extremal metric,
providing an important necessary condition for Sasaki-extremal metrics. Another
application is a proof of the uniqueness of Sasaki-extremal metrics, for a
fixed transversally holomorphic structure on the Reeb foliation.
