Inequalities : Theory of majorization and its applications

insisting that the equality sign holds when k = n. Here, x[X] > • • • > x[n] are the xt arranged in decreasing order and, similarly, y[X] > • • • > y[n]. If (1) is only required for the increasing (decreasing) convex functions on R then one speaks of weak sub-majorization x >, respectively). The first is equivalent to (2). Let & be an open convex subset of R which is symmetric, that is, invariant under each permutation of the coordinates. A function : & -> R is said to be Schur increasing (or Schur convex) if it is nondecreasing relative to the partial ordering x < y of & ; similarly for Schur decreasing functions, also called Schur concave functions. A Schur increasing function is always symmetric. An obvious example would be