I will explain my proof of the inequality $c_1^2\leq 5c_2$ for a class of smooth, projective surfaces over algebraically closed fields of characteristic $p>0$. My approach is based on a study of slopes of Frobenius morphism on crystalline cohomology of $X$ and of the de Rham-Witt complex of $X$. In particular my methods do not require any lifting hypothesis.