Entropy-type inequalities for generalized Gamma densities

We investigate the relaxation to equilibrium of the solution of a class of one-dimensional linear Fokker-Planck type equations that have been recently considered in connection with the study of addiction phenomena in a system of individuals. The steady states of these equations belong to the class of generalized Gamma densities. As a by-product of the relaxation analysis, we prove new weighted Poincaré and logarithmic Sobolev type inequalities for this class of densities.