Convex cones of generalized multiply monotone functions and the dual cones

Abstract

Let $n$ and $k$ be nonnegative integers such that $1\le k\le n+1$. The convex cone ${\mathcal{F}}_{+}^{k:n}$ of all functions $f$ on an arbitrary interval $I\subseteq \mathbb{R}$ whose derivatives $f^{(j)}$ of orders $j=k-1,\dots ,n$ are nondecreasing is characterized in terms of extreme rays of the cone ${\mathcal{F}}_{+}^{k:n}$. A simple description of the convex cone dual to ${\mathcal{F}}_{+}^{k:n}$ is given. 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 $j$th order in place of $f^{(j)}$. Somewhat similar results were previously obtained 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.

Version 2: More applications given; two typos fixed

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