  Consider the blow up $\pi: \widetilde{X} \to X$ of a rational surface $X$ at
a point. Let $\widetilde{V}$ be a holomorphic bundle over $\widetilde{X}$ whose
restriction to the exceptional divisor equals ${\cal{O}(j) \oplus {\cal O}(-j)$
and define $V =(\pi_*\widetilde{V})^{\vee \vee}.$ Friedman and Morgan gave the
following bounds for the second Chern classes $j \leq c_2(\widetilde{V}) -
c_2(V) \leq j^2.$ We show that these bounds are sharp.
