Pade approximants of the Mittag-Leffler functions

It is shown that for m <= n the Pade approximants {pi(n,m)(center dot;F-gamma)}, which locally deliver the best rational approximations to the Mittag-Leffler functions F-gamma, approximate the F-gamma as n -> infinity uniformly on the compact set D = {z : |z| <= 1} at a rate asymptotically equal to the best possible one. In particular, analogues of the well-known results of Braess and Trefethen relating to the approximation of exp z are proved for the Mittag-Leffler functions.