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 ƒ* of a function ƒ. If ƒ(x) is a.e. differentiable on [0, 1], Φ is a nonnegative Borel function on R, and Ψ: [0, ∞) → R is increasing then ∫10Φ (ƒ*) Ψ(|ƒ*)′|) ≤ ∫10Φ(ƒ) Ψ(|ƒ′|). Furthermore, if Φ is strictly positive, Ψ is strictly increasing, ƒ is absolutely continuous, and there is equality above with both integrals being finite then ƒ must be monotone.