  In this paper, we consider the curvature strict positivity of direct image
bundles (vector bundles) associated to a strictly pseudoconvex family of
bounded domains. The main result is that the curvature of the direct image
bundle (vector bundles) associated to a strictly pseudoconvex family of bounded
domains is strictly positive in the sense of Nakano even if the curvature of
the original vector bundle is just Nakano positive. Based on our (I and my
coauthors) previous results, this result further demonstrates that strictly
pseudoconvex domains and pseudoconvex domains have very different geometric
properties. To consider the curvature strict positivity, we will first
construct the curvature formulas for vector bundles, then the curvature strict
positivity of a strictly pseudoconvex family of bounded domains will be a
simple consequence of it. As applications to convex analysis, we get a
corresponding version of Pr\'ekopa's Theorem, i.e., we get the strict
positivity of a strictly convex family of bounded domains.
