  It is known that real Non-Gaussian stable distributions are unimodal, not
additive strongly unimodal, and multiplicative strongly unimodal in the
symmetric case. By a theorem of Cuculescu-Theodorescu, the only remaining
relevant situation for the multiplicative strong unimodality of stable laws is
the one-sided. In this paper, we show that positive $\alpha-$stable laws are
multiplicative strongly unimodal iff $\alpha\le 1/2.$
