Slope inequalities and a Miyaoka-Yau type inequality

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