  Let ${\bf L}$ be the unit exponential random variable and ${\bf Z}_\alpha$
the standard positive $\alpha$-stable random variable. We prove that
$\{(1-\alpha) \alpha^{\gamma_\alpha} {\bf Z}_\alpha^{-\gamma_\alpha}, 0< \alpha
<1\}$ is decreasing for the optimal stochastic order and that $\{(1-\alpha){\bf
Z}_\alpha^{-\gamma_\alpha}, 0< \alpha < 1\}$ is increasing for the convex
order, with $\gamma_\alpha = \alpha/(1-\alpha).$ We also show that
$\{\Gamma(1+\alpha) {\bf Z}_\alpha^{-\alpha}, 1/2\le \alpha \le 1\}$ is
decreasing for the convex order, that ${\bf Z}_\alpha^{-\alpha}\,\prec_{st}\,
\Gamma(1-\alpha) \L$ and that $\Gamma(1+\a){\bf Z}_\alpha^{-\alpha}
\,\prec_{cx}\,{\bf L}.$ This allows to compare ${\bf Z}_\alpha$ with the two
extremal Fr\'echet distributions corresponding to the behaviour of its density
at zero and at infinity. We also discuss the applications of these bounds to
the strange behaviour of the median of ${\bf Z}_\alpha$ and ${\bf
Z}_\alpha^{-\alpha}$ and to some uniform estimates on the classical
Mittag-Leffler function. Along the way, we obtain a canonical factorization of
${\bf Z}_\alpha$ for $\alpha$ rational in terms of Beta random variables. The
latter extends to the one-sided branches of real strictly stable densities.
