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Strict inequalities for integrals of decreasingly rearranged functions*

  • Avner Friedman (a1) and Bryce McLeod (a2)

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.

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2 J. E. Brothers . Integral geometry in homogeneous spaces. Trans. Amer. Math. Soc. 124 (1966), 480517.

8 G. Pólya and G. Szegö . Isoperimetric Inequalities in Mathematical Physics (Princeton, N.J.: Princeton University Press, 1951).

10 A. Sard . The measure of the critical values of differentiable maps. Bull. Amer. Math. Soc. 48 (1942), 883890.

12 E. Sperner . Zur Symmetrisierung von Funktionen auf Sphären. Math. Z. 134 (1973), 317–327.

13 G. Talenti . Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110 (1976), 353–372.

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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