  In the first part of the paper, we study a Fujita-type conjecture by Popa and
Schnell, and give an effective bound on the generic global generation of the
direct image of the twisted pluricanonical bundle. We also point out the
relation between the Seshadri constant and the optimal bound. In the second
part, we give an affirmative answer to a question by
Demailly-Peternell-Schneider in a more general setting. As an application, we
generalize the theorems by Fujino and Gongyo on images of weak Fano manifolds
to the Kawamata log terminal cases, and refine a result by Broustet and
Pacienza on the rational connectedness of the image.
