Banach Journal of Mathematical Analysis

Convex cones of generalized multiply monotone functions and the dual cones

Iosif Pinelis

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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

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

Received: 22 July 2015
Accepted: 1 February 2016
First available in Project Euclid: 7 October 2016

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Zentralblatt MATH identifier

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]

dual cones multiply monotone functions generalized moments stochastic orders probability inequalities


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.

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