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

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