Let $f: X \to B$ be a relatively minimal fibration of maximal Albanese dimension from a variety X of dimension $n \ge 2$ to a curve B defined over an algebraically closed field of characteristic zero. We prove that $K_{X/B}^n \ge 2n! \chi _f$ . It verifies a conjectural formulation of Barja in [2]. Via the strategy outlined in [4], it also leads to a new proof of the Severi inequality for varieties of maximal Albanese dimension. Moreover, when the equality holds and $\chi _f> 0$ , we prove that the general fibre F of f has to satisfy the Severi equality that $K_F^{n-1} = 2(n-1)! \chi (F, \omega _F)$ . We also prove some sharper results of the same type under extra assumptions.