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Strict inequalities for integrals of decreasingly rearranged functions*

Published online by Cambridge University Press:  14 November 2011

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

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
Copyright
Copyright © Royal Society of Edinburgh 1986

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