## Banach Journal of Mathematical Analysis

### Convex cones of generalized multiply monotone functions and the dual cones

Iosif Pinelis

#### Abstract

Let $n$ and $k$ be nonnegative integers such that $1\le k\leq n+1$. The convex cone $\mathscr{F}_{+}^{k:n}$ 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. A simple description of the convex cone dual to $\mathscr{F}_{+}^{k:n}$ 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 $f^{(j)}$. 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 difficulties.

#### Article information

Source
Banach J. Math. Anal. Volume 10, Number 4 (2016), 864-897.

Dates
Accepted: 1 February 2016
First available in Project Euclid: 7 October 2016

http://projecteuclid.org/euclid.bjma/1475870136

Digital Object Identifier
doi:10.1215/17358787-3649788

Zentralblatt MATH identifier
06667684

#### Citation

Pinelis, Iosif. Convex cones of generalized multiply monotone functions and the dual cones. Banach J. Math. Anal. 10 (2016), no. 4, 864--897. doi:10.1215/17358787-3649788. http://projecteuclid.org/euclid.bjma/1475870136.

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