Positivity of anticanonical divisors in algebraic fibre spaces

Let $f:X\to Y$ be an algebraic fibre space between normal projective varieties and $F$ be a general fibre of $f$. We prove an Iitaka-type inequality $\kappa (X,-K_X)\le \kappa (F,-K_F)+\kappa (Y,-K_Y)$ under some mild conditions. We also obtain some more results relates the positivity of $-K_X$ and $-K_Y$.