Comparing Fréchet and positive stable laws

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