[hal-00458257, v1] On the unimodality of inverse positive stable laws

We observe that the function Fα(x) = (1+αx α)e−x α is completely monotone iff α ≤ α0 for some α0 ∈ ]2/3, 3/4[. This property is equivalent to the unimodality of the inverse positive α-stable law. The random variable associated with Fα appears then in two different factorizations of the positive α-stable distribution. Furthermore, it is infinitely divisible iff α ≤ α1 for some α1 ∈ ]2/3, α0[ and self-decomposable iff α ≤ α2 for some α2 ∈ ]2/3, α1[. The purpose of this note is to point out a curious phenomenon for positive stable distributions for which we did not find any trace in the literature, though it could be of some interest. For every α ∈ ]0, 1[, let fα be the positive α−stable density and Zα the corresponding random variable, normalized such that