alpha-stable densities are HCM for alpha \in ]0,1/4] and [1/3,1/2]

We investigate the problem raised by L. Bondesson, about the hyperbolic complete monotonicity of $\alpha$-stable densities. We prove that densitites of subordinators of order $\alpha$ are HCM for $\alpha \in ]0,1/4] \cup [1/3,1/2]$.