.  \[x-thm00.0\] Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$ onto a smooth projective variety $Y$ with connected fibers. Let $\Delta$ be an effective $\mathbb R$-divisor on $X$ such that $K_X+\Delta$ is $\mathbb R$-Cartier and that $(X, \Delta)$ is log canonical over a nonempty Zariski open set of $Y$. Let $D$ be an $\mathbb R$-Cartier $\mathbb R$-divisor on $X$ such that $D-(K_{X/Y}+\Delta)$ is nef. Then, for any $\mathbb R$-divisor $Q$ on $Y$, we have $$\kappa _\sigma(X, D+f^*Q)\geq \kappa _\sigma(F, D|_F) +\kappa (Y, Q)$$ and $$\kappa _\sigma(X, D+f^*Q)\geq \kappa (F, D|_F) +\kappa _\sigma(Y, Q)$$ where $F$ is a sufficiently general fiber of $f:X\to Y$.  