On the unimodality of power transformations of positive stable densities

Let $Z_{\alpha }$ be a positive $\alpha -$stable random variable and $r\in{\bf R}.$Weshowtheexistenceofanunboundedopendomain$D$in$[1/2,1]\times{\bf R}$withacuspat$(1/2,-1/2)$, characterized by the complete monotonicity of the function $F_{\alpha, r} (\lambda) = (\alpha \lambda^\alpha -r)e^{-\lambda^\alpha}/!/! ,$suchthat$Z_\alpha^r$ is unimodal if and only if $(\alpha ,r)\mathrm{\notin }D\mathrm{.}$