  L.Bondesson [1] conjectured that the density of a positive $\alpha$-stable
distribution is hyperbolically completely monotone (HCM in short) if and only
if $\alpha$ $\le$ 1/2. This was proved recently by P. Bosch and Th. Simon, who
also conjectured a strengthened version of this result. We disprove this
conjecture as well as a correlated conjecture of Bondesson, while giving a
short new proof of the initial conjecture, as a direct consequence of a new
algebraic property of HCM and Generalized Gamma convolution densities (GGC in
short) which we establish.
