Comparing Frechet and positive stable laws

Let L be the unit exponential random variable and Z(alpha) the standard positive alpha-stable random variable. We prove that {(1-alpha)alpha(gamma alpha)Z(alpha)(-gamma alpha), 0 < alpha < 1} is decreasing for the optimal stochastic order and that {(1-alpha)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)Z(alpha)(-alpha); 1/2 <= alpha <= 1} is decreasing for the convex order, that Z(alpha)(-alpha) <(st) Gamma (1-alpha)L and that Gamma(1+alpha)Z(alpha)(-alpha) <(cx) L. This allows one to compare Z(alpha) with the two extremal Frechet 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 Z(alpha) and Z(alpha)(-alpha) and to some uniform estimates on the classical Mittag-Leffler function. Along the way, we obtain a canonical factorization of Z(alpha) for alpha rational in terms of Beta random variables. The latter extends to the one-sided branches of real strictly stable densities.