On the unimodality of power transformations of positive stable densities

Let $Z_\alpha$ be a positive $\alpha-$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 $F_{\alpha, r} (\lambda) = (\alpha \lambda^\alpha -r)e^{-\lambda^\alpha}/!/! ,$ such that $Z_\alpha^r$ is unimodal if and only if $(\alpha, r)\notin D.$