Journal of Inequalities and Applications 
Volume 5 (2000), Issue 4, Pages 309-320
doi:10.1155/S1025583400000151

A general rearrangement inequality à la Hardy–Littlewood

Friedemann Brock

 Universität Leipzig, Fakultät für Mathematik und Informatik,, Augustusplatz 10, Leipzig D-04109, Germany
Received 5 May 1999; Revised 12 July 1999

Abstract

Let F=F(v1,…,vm) smooth on (R0+)m with Fvivj≥0 for i≠j. Furthermore, let u1,…,um nonnegative and bounded functions on Rn with compact support. We prove the inequality ∫RnF(u1,⋯,um)dx≤∫RnF(u1∗,⋯,um∗)dx, where ∗ denotes symmetric decreasing rearrangement.