  For a minimal smooth projective surface $S$ of general type over a field of
characteristic $p>0$, we prove that $K^2_S\le 32\chi(\cal{O}_S).$ Moreover, if
$18\chi(\cal{O}_S)<K^2_S\le 32\chi(\cal{O}_S)$, Albanese morphism of $S$ must
induces a genus two fiberation. A classification of surfaces with
$K^2_S=32\chi(\cal{O}_S)$ is also given. The inequality also implies
$\chi(\cal{O}_S)>0$, which answers completely a question of Shepherd-Barron.
