The Noether inequalities for a foliated surface of general type

Let (F,S) be a foliated surface of general type with reduced singularities over the complex number. We establish the Noether type inequalities for (F,S). Namely, we prove that vol(F) ≥ pg(F)-2, and that vol(F) ≥ 2pg(F)-4 if moreover the surface S is also of general type. Examples show that both of the Noether type inequalities are sharp.