Schur-Ostrowski Theorems for Functionals on L1(0,1).
Abstract
Hardy, Littlewood and Polya (1934) introduced the partial ordering of majorization among n-dimensional real vectors. Many well known inequalities can be recast as the statement that certain functions are increasing with respect to this ordering. Such functions are said to be Schur-convex. An important result in the theory of majorization is the Shur-Ostrowski Theorem, which characterizes Schur-convex functions. The concept of majorization has been extended to elements of L sub 1(0,1) by Ryff (1963). A functional on L sub 1(0,1) that is increasing with respect to the ordering of majorization is said to be Schur-convex. In this paper, the authors prove an analogue of the Schur-Ostrowski condition which characterizes Schur-convex functionals in terms of their Gateaux differentials. They also introduce another partial ordering in L sub 1(0,1) called unrestricted majorization. This partial ordering is similar to majorization but does not involve the use of decreasing rearrangements. The authors establish a characterization of non-decreasing functionals on L1(0,1) with respect to the partial ordering of unrestricted majorization through another analogue of the Schur-Ostrowski condition. Keywords include: Inequalities; majorization; Muirhead's theorem; peakedness in symmetric distribution; rearrangement; Schur functions; Schur-Ostrowski's theorem.
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 01, 1984
- Accession Number
- ADA150193
Entities
People
- Frank Proschan
- Jayaram Sethuraman
- Warren Chan
Organizations
- Florida State University