Open Access
October 2016 Convex cones of generalized multiply monotone functions and the dual cones
Iosif Pinelis
Banach J. Math. Anal. 10(4): 864-897 (October 2016). DOI: 10.1215/17358787-3649788


Let n and k be nonnegative integers such that 1kn+1. The convex cone F+k:n of all functions f on an arbitrary interval IR whose derivatives f(j) of orders j=k1,,n are nondecreasing is characterized. A simple description of the convex cone dual to 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 jth 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.


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Iosif Pinelis. "Convex cones of generalized multiply monotone functions and the dual cones." Banach J. Math. Anal. 10 (4) 864 - 897, October 2016.


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

zbMATH: 1364.90264
MathSciNet: MR3555754
Digital Object Identifier: 10.1215/17358787-3649788

Primary: 46N10
Secondary: 26A46 , 26A48 , 26A51‎ , 26D05 , 26D07 , 26D10 , 26D15 , 34L30 , 41A10 , 46A55 , 49K30 , 49M29 , 52A07 , 52A41 , 60E15 , 90C25 , 90C46

Keywords: dual cones , Generalized moments , multiply monotone functions , Probability inequalities , Stochastic orders

Rights: Copyright © 2016 Tusi Mathematical Research Group

Vol.10 • No. 4 • October 2016
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