Let ƒ, g be measurable non-negative functions on R , and let \ ̄ tf, ḡ be their equimeasurable symmetric decreasing rearrangements. Let F : R × R → R be continuous and suppose that the associated rectangle function defined by F(R) = F(a, c) + F(b, d) − F(a, d) − F(b, c) for R = [(x, y) ϵ R 2 ¦ a ⩽ x ⩽ b, c ⩽ y ⩽ d] , is non-negative. Then ∝ F(ƒ, g) dμ ⩽ ∝ F( \ ̄ tf, g ̄ ) dμ , where μ is Lebesgue measure. The concept of equimeasurable rearrangement is also defined for functions on a more general class of measure spaces, and the inequality holds in the general case. If F ( x , y ) = − ϑ ( x − y ), where ϑ is convex and ϑ(0) = 0, then we obtain ∝ ϑ( \ ̄ tf − g ̄ ) dμ ⩽ ∝ ϑ(ƒ − g) dμ . In particular, if ϑ(x) = ¦x¦ p , 1 ⩽ p ⩽ +∞ , then we find that the operator S:ƒ → \ ̄ tf is a contraction on L p for 1 ⩽ p ⩽ +∞.