## Banach Journal of Mathematical Analysis

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

### 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.

#### Article information

**Source**

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

**Dates**

Received: 22 July 2015

Accepted: 1 February 2016

First available in Project Euclid: 7 October 2016

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/17358787-3649788

**Zentralblatt MATH identifier**

06667684

**Subjects**

Primary: 46N10: Applications in optimization, convex analysis, mathematical programming, economics

Secondary: 26A48: Monotonic functions, generalizations 26A51: Convexity, generalizations 26A46: Absolutely continuous functions 26D05: Inequalities for trigonometric functions and polynomials 26D07: Inequalities involving other types of functions 26D10: Inequalities involving derivatives and differential and integral operators 26D15: Inequalities for sums, series and integrals 34L30: Nonlinear ordinary differential operators 41A10: Approximation by polynomials {For approximation by trigonometric polynomials, see 42A10} 46A55: Convex sets in topological linear spaces; Choquet theory [See also 52A07] 49K30: Optimal solutions belonging to restricted classes 49M29: Methods involving duality 52A07: Convex sets in topological vector spaces [See also 46A55] 52A41: Convex functions and convex programs [See also 26B25, 90C25] 60E15: Inequalities; stochastic orderings 90C25: Convex programming 90C46: Optimality conditions, duality [See also 49N15]

**Keywords**

dual cones multiply monotone functions generalized moments stochastic orders probability inequalities

#### 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. https://projecteuclid.org/euclid.bjma/1475870136