On a conjecture about the Frank copula family

It is shown that any member of the Frank copula family fulfills a family of functional inequalities which arises as a family of sufficient conditions that allow to characterize the transitivity of reciprocal relations expressing the winning probabilities among random variables artificially coupled by a same copula. The inequalities are parameterized by a single integer parameter. The first of these inequalities expresses symmetry of the copula, while the second one expresses an upper bound on its volume. Subsequent inequalities are increasingly restrictive.