On algebraic surfaces of general type with negative c2

We prove that for any prime number $p\ge 3$, there exists a positive number ${\kappa }_p$ such that $\chi ({\mathcal{O}}_X)\ge {\kappa }_p{c}_1^2$ holds true for all algebraic surfaces $X$ of general type in characteristic $p$. In particular, $\chi ({\mathcal{O}}_X)>0$. This answers a question of N. Shepherd-Barron when $p\ge 3$.