  By using a general version of curvature condition, derivative inequalities
are established for a large class of subelliptic diffusion semigroups. As
applications, the Harnack/cost-entropy/cost-variance inequalities for the
diffusion semigroups, and the Poincar\'e/log-Sobolev inequalities for the
associated Dirichlet forms in the symmetric case, are derived. Our results
largely generalize and partly improve the corresponding ones obtained recently
in [BB].
