On the unimodality of power transformations of positive stable densities

Let Z_α be a positive α‐stable random variable and $rin {mathbb {R}}.Let Z_α be a positive α‐stable random variable and $rin {mathbb {R}}.$ We show the existence of an unbounded open domain D in [1/2, 1] × ( ? ∞, ?1/2] 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 (α, r)?D.