An Integral Inequality for Derivatives of Equimeasurable Rearrangements

Let f be a Lebesgue measurable function on I = [0, 1] which is finite-valued almost everywhere and let f* be the nonincreasing rearrangement of f. In Section 2 we shall study variation reducing properties of the operator f ↦ f* and how continuity of f reflects on that of f*. In Section 3 we shall prove ∫10F(f*(ζ), |f*′(ζ)|) dζ ≤ ∫10F(f(x), |f′(x)|)dx, where f is almost everywhere differentiable and F(y1,y2) is a Borel measurable function on R × [0, + ∞) such that F(y1, y2) is a nondecreasing function of y2 for each fixed y1. We shall also discuss when the equality holds in the inequality.