Vector joint majorization and generalization of Csiszár-Körner's inequality for f-divergence

In this paper we study vector joint majorization for comparing m-tuple and n-tuple of pairs in the Cartesian product of two real linear spaces. We show a Sherman type inequality for a vector-valued ≤C-convex function f, where ≤C is a cone ordering. We also prove a Hardy-Littlewood-Pólya-Karamata type theorem for f. As applications, we generalize Csiszár-Körner's inequality for f-divergence of two n-tuples of positive numbers. In doing so, we employ n-tuples of pairs of positive linear maps and positive operators on a Hilbert space.