Jensen’s inequality for spectral order and submajorization ✩

LetAbe a C∗-algebra and φ :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 ∈ A whose spectrum lies in I , we obtain a Jensen’s-type inequality f (φ(a)) φ(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. © 2006 Elsevier Inc. All rights reserved