Poincaré-type inequalities for stable densities

In a recent paper we introduced the concept of fractional score, a generalization of the linear score function, well-known in theoretical statistics. As the Gaussian density is closely related to the linear score, the fractional score function allows to identify Lévy stable laws as the (unique) probability densities for which the score of a random variable X is proportional to - X. We use this analogy to extend to stable laws the classical Poincaré inequality for Gaussian densities. Application of this inequality allows to obtain bounds on moments of stable laws, and a sharp one-dimensional version of Hardy-Poincaré inequality.