The Frank inequality

We investigate a functional inequality for copulas that has emerged from our study of the comparison of a set of random variables pairwisely coupled by a same copula. Any copula satisfying this inequality is necessarily symmetric and radially symmetric. Moreover, any associative copula satisfying this inequality is a solution to the well-known Frank equation. For this reason, the inequality is coined the Frank inequality. We fully characterize the associative copulas that satisfy the Frank inequality: they turn out to be either Frank copulas or ordinal sums of a same Frank copula with equidistant idempotent elements. As a by-product, we observe that Frank copulas are super-additive on the unit square.