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Strict inequalities for integrals of decreasingly rearranged functions*

Published online by Cambridge University Press:  14 November 2011

Avner Friedman  and
Bryce McLeod
Avner Friedman
Affiliation:
Northwestern University, Evanston, Illinois, U.S.A.
Bryce McLeod
Affiliation:
Oxford University, Mathematical Institute, Oxford, England
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Synopsis

It is well known that if f, g, h are nonnegative functions and f*, g*, h* their symmetrically decreasing rearrangements, then

also if u* is a spherically decreasing rearrangement of a function u,

In this paper it is proved under suitable assumptions (including the assumption that h is already rearranged) that equality holds in (i) if and only if f and g are already rearranged, and, if 1 < p < ∞ equality holds in (ii) if and only if u is already rearranged. We discuss (ii) both in ℝn and on the unit sphere.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 102 , Issue 3-4 , 1986 , pp. 277 - 289
Copyright
Copyright © Royal Society of Edinburgh 1986

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