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Functional Inequalities related to the Rogers-Shephard Inequality

Published online by Cambridge University Press:  21 December 2009

Andrea Colesanti
Andrea Colesanti
Affiliation:
Dipartimento di Matematica “U. Dini”, viale Morgagni 67/A, 50134 Firenze, Italy. E-mail: colesant@math.unifi.it
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Abstract

In this paper a notion of difference function Δf is introduced for real-valued, non-negative and log-concave functions f defined in Rn. The difference function represents a functional analogue of the difference body K + (−K) of a convex body K. The main result is a sharp inequality which bounds the integral of Δf from above in terms of the integral of f. Equality conditions are characterized. The investigation is extended to an analogous notion of difference function for α-concave functions, with α < 0. In this case also an upper bound for the integral of the α-difference function of f in terms of the integral of f is proved. The bound is sharp in the case α = −∞ and in the one-dimensional case.

Type
Research Article
Information
Mathematika , Volume 53 , Issue 1 , June 2006 , pp. 81 - 101
Copyright
Copyright © University College London 2006

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