Convex cones of generalized multiply monotone functions and the dual cones

Document Type

Article

Publication Date

1-1-2016

Abstract

© 2016 by the Tusi Mathematical Research Group. Let n and k be nonnegative integers such that 1 ≤ k ≤ n + 1. The convex cone F+k:n of all functions f on an arbitrary interval I ⊆ ℝ whose derivatives f(j) of orders j = k - 1;:::; n are nondecreasing is characterized. A simple description of the convex cone dual to F+k:n is given. In particular, these results are useful in, and were motivated by, applications in probability. In fact, the results are obtained in a more general setting with certain generalized derivatives of f of the jth order in place of f(j). Somewhat similar results were previously obtained, in terms of Tchebycheff-Markov systems, in the case when the left endpoint of the interval I is finite, with certain additional integrability conditions; such conditions fail to hold in the mentioned applications. Development of substantially new methods was needed to overcome the diffculties.

Publication Title

Banach Journal of Mathematical Analysis

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