On the unimodality of power transformations of positive stable densities

Abstract

Let $Z_α$ be a positive $α-$stable random variable and $r\in{\bf R}.$ We show the existence of an unbounded open domain $D$ in $[1/2,1]\times{\bf R}$ with a cusp at $(1/2,-1/2)$, characterized by the complete monotonicity of the function $F_{α, r} (λ) = (αλ^α-r)e^{-λ^α}/!/! ,$ such that $Z_α^r$ is unimodal if and only if $(α, r)\notin D.$