Convexity inequalities for positive operators

We prove pointwise convexity (Jensen-type) inequalities of the form F(Tf) <= T[F(f)] where F is a convex function defined on a convex subset of some Banach space X and T is the X-valued extension of a positive operator on some function space. Examples include the pointwise Holder inequality T(fg) <= (Tf-p)(1/p) (Tf-q)(1/q) for a positive sublinear operator T. As applications we consider vector-valued conditional expectation and a "real" proof of the Riesz-Thorin theorem for positive operators.