This article shows a number of strong inequalities that hold for the Chern numbers
c
1
2
c_1^2
,
c
2
c_2
of any ample vector bundle
E
\mathcal {E}
of rank
r
r
on a smooth toric projective surface,
S
S
, whose topological Euler characteristic is
e
(
S
)
e(S)
. One general lower bound for
c
1
2
c_1^2
proven in this article has leading term
(
4
r
+
2
)
e
(
S
)
ln
2
(
e
(
S
)
12
)
(4r+2)e(S)\ln _2\left (\tfrac {e(S)}{12}\right )
. Using Bogomolov instability, strong lower bounds for
c
2
c_2
are also given. Using the new inequalities, the exceptions to the lower bounds
c
1
2
>
4
e
(
S
)
c_1^2> 4e(S)
and
c
2
>
e
(
S
)
c_2>e(S)
are classified.