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 Hölder 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.