  This paper investigates the curvature properties of higher direct images $
R^qf_*\Omega_{X/S}^p(E)$, where $f: X\rightarrow S$ is a family of compact
K\"ahler manifolds equipped with a hermitian vector bundle $E \rightarrow X$.
We derive a general curvature formula and explore several special cases,
including those where $p + q = n$, $q = 0$, and $p = n$, with $E$ being a line
bundle. Furthermore, the paper examines the curvature in the context of
fiberwise hermitian flat cases, families of Hermite-Einstein vector bundles,
and applications to moduli spaces and Weil-Petersson metrics, providing some
insight into their geometric and analytical properties.
