Let ∼≺ be a partial ordering
among the points of a set D ⊂ Rn. A real-valued function f defined on D is said to
preserve ≺∼ if x,y ∈ D,x ≺∼ y implies f(x) ≦ f(y). The central theorem of this
paper gives necessary and sufficient conditions for f 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 − x ∈ C. Corollaries to the theorem consider the case when
f is differentiable and ∼≺ is order isomorphic to a cone ordering under
a differentiable 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.
