Jensen's inequality for spectral order and submajorization

Let A be aC*-algebra and phi: A -> L(H) be a positive unital map. Then, for a convex function f : I -> R defined on some open interval and a self-adjoint element a E A whose spectrum lies in I, we obtain a Jensen's-type inequality f (phi (a)) <= phi (f (a)) where <= denotes an operator preorder (usual order, spectral preorder, majorization) and depends on the class of convex functions considered, i.e., monotone convex or arbitrary convex functions. Some extensions of Jensen's-type inequalities to the multi-variable case are considered. (c) 2006 Elsevier Inc. All rights reserved.