INTEGRAL-INEQUALITIES AND EQUALITIES FOR THE REARRANGEMENT OF HARDY AND LITTLEWOOD

The main purpose of this paper is to prove the following result about the Hardy-Littlewood decreasing rearrangement f of a function f. If f (x) is a.e. differentiable on [0, 1], PHI is a nonnegative Borel funct ion on R, and PSI: [0, infinity) --> R is increasing then intergral-1/ 0 PHI (f) PSI(\f*)'\less-than-or-equal-to intergral-1/0 PHI(f) PSI(\f '\). Furthermore, if PHI is strictly positive, PSI is strictly increas ing, f is absolutely continuous, and there is equality above with both integrals being finite then f must be monotone. (C) 1994 Academic Pre ss, Inc.