Order-preserving functions: Applications to majorization and order statistics
Albert W. Marshall, David W. Walkup, Roger J.‐B. Wets · Pacific Journal of Mathematics · 1967
Let < be a partial ordering among the points of a set D c R n . A real-valued function / defined on D is said to preserve ^ if x,yeD, x^y implies f(x) ^ f(y).The central theorem of this paper gives necessary and sufficient conditions for / to preserve ^ if ^ is a cone ordering, i.e. if there exists a convex cone C such that x ;< y if and only if y -xeC.Corollaries to the theorem consider the case when / is differentiable and ;< is order isomorphic to a cone ordering under a differentiate mapping.It is seen that the ordering of majorization is a special case of a cone ordering and that a straightforward application of a corollary yields the results of Schur and Ostrowski on functions which preserve majorization.The corollaries are also applied to a partial ordering of positive semi-definite matrices and to certain partial orderings arising in the theory of order statistics.