  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 $K_{X/B}^n
\ge 2n! \chi_f$, which was conjectured by Barja in [2]. Via the strategy
outlined in [5], 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 fiber $F$ 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.
