Vol. 23, No. 3, 1967

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Order-preserving functions: Applications to majorization and order statistics

Albert W. Marshall, David William Walkup and Roger Jean-Baptiste Robert Wets

Vol. 23 (1967), No. 3, 569–584

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.

Mathematical Subject Classification
Primary: 26.50
Secondary: 15.00
Received: 9 January 1967
Published: 1 December 1967
Albert W. Marshall
David William Walkup
Roger Jean-Baptiste Robert Wets