Positivity of anticanonical divisors in algebraic fibre spaces

Let $f:X\rightarrow Y$ be an algebraic fibre space between normal projective varieties and $F$ be a general fibre of $f$. We prove an Iitaka-type inequality $\kappa(X,-K_X)\leq \kappa(F,-K_F)+\kappa(Y,-K_Y)$ under some mild conditions. We also obtain some more results relates the positivity of $-K_X$ and $-K_Y$.Comment: 25 pages. Revise due to a gap found after the article was published. The result of the main theorem (Theorem 1.1 and Theorem 4.1) is intact, but Theorem 3.8, Proposition 4.2, and Theorem 4.3 hold only under some assumptions stronger than the previous version. Also, we give an affirmative answer to Question 5.9 in this versio