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 $f^{\left(j\right)}$ of orders $j=k-1,\dots ,n$ are nondecreasing is characterized. A simple description of the convex cone dual to ${\mathcal{F}}_{+}^{k:n}$ 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 $j$th order in place of $f^{\left(j\right)}$. 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 difficulties.