Chern numbers of certain Lefschetz fibrations

We address the geography problem of relatively minimal Lefschetz fibrations over surfaces of nonzero genus and prove that if the fiber-genus of the fibration is positive, then 0 less than or equal to c(1)(2) less than or equal to 5c(2) (equivalently, 0 less than or equal to c(1)(2) less than or equal to 10 chi(h)) holds for those symplectic 4-manifolds. A useful characterization of minimality of such symplectic 4-manifolds is also proved.