Rational approximation of the Mittag-Leffler functions

It is shown that for $m-1\le n$ the Padé approximants $\{\pi_{n,m}(\cdot;F_\gamma)\}$, which locally deliver the best rational approximations to the Mittag-Leffler functions $F_\gamma$, approximate the $F_\gamma$ as $n\to\infty$ uniformly on the compact set $D=\{z:|z|\le1\}$ at a rate asymptotically equal to the best possible one. In particular, analogues of the well-know results of Braess and Trefethen relating to the approximation of $\exp(z)$ are proved for the Mittag-Leffler functions.