X\to Y$ be a surjective morphism between smooth projective varieties with $\dim Y=n$. Let $\mathcal L$ be any ample invertible sheaf on $Y$. Then, for every positive integer $k$, the sheaf $$f_*\omega^{\otimes k}_{X/Y}\otimes  \omega_Y\otimes \mathcal L^{\otimes l}$$ is generated by global sections for $l\geq 