Multiplicative strong unimodality for positive stable laws

Abstract: 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 $\alpha\le 1/2.$