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 $\chi(\mathcal{O}_X)\ge \kappa_pc_1^2$ 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 $p\ge 3$.