An inequality of rearrangements on the unit circle

Abstract

We prove that the integral of the product of two functions over a symmetric set in $\mathbb{S}^1\times\mathbb{S}^1$, defined as $E=\{(x,y)\in\mathbb{S}^1\times\mathbb{S}^1:d(\sigma_1(x),\sigma_2(y))\leq\alpha\}$ (where $\sigma_1$, $\sigma_2$ are diffeomorphisms of $\mathbb{S}^1$ with certain properties and $d$ is the geodesic distance on $\mathbb{S}^1$), increases when we pass to their symmetric decreasing rearrangement. We also give a characterization of the diffeomorphisms $\sigma_1$, $\sigma_2$ for which the rearrangement inequality holds. As a consequence, we obtain the result for the integral of the function $\varPsi(f(x),g(y))$ (where $\varPsi$ is a supermodular function) with a kernel given as $k[d(\sigma_1(x),\sigma_2(y))]$, with $k$ decreasing.