  We shall show that $q$-semipositivity of the vector bundle $E$ over a
K\"ahler total space $\mathcal X$ implies the Griffiths-semipositivity of the
$q$-th direct image of $\mathcal O(K_{\mathcal X/B}\otimes E)$. As an
application, we shall give a negative-curvature criterion for the generalized
Weil-Petersson metric on the base manifold.
