For any given partial ordering \(\preceq\)of a setX, real-valued functionsf1 defined onXwhich satisfyf(x)= f(y) wheneverx_yare var- 2 iously referred to as “monotonic,” “isotonic,” or “order-preserving.” 3 For the ordering of majorization, the order-preserving functions were 4 first systematically studied by I. Schur (1923). In Schur’s honor, such 5 functions are said to be “convex in the sense of Schur,” “Schur-convex,” 6 or “S-convex.” The historical origin of these terms is described in 7 Section 1.C.
KeywordsConvex Cone Total Positivity Semigroup Property Elementary Symmetric Function Decrease Hazard Rate
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