  Let $\sE$ be an ample rank $r$ bundle on a smooth toric projective surface,
$S$, whose topological Euler characteristic is $e(S)$. In this article, we
prove a number of surprisingly strong lower bounds for $c_1(\sE)^2$ and
$c_2(\sE)$. We also enumerate the exceptions to either the inequality
$c_1(\sE)^2\ge 4e(S)$ or the inequality $c_2(\sE)\ge e(S)$ holding.
