Curvature positivity of invariant direct images of Hermitian vector bundles

We prove that the invariant part, with respect to a compact group action satisfying certain condition, of direct image Nakano positive Hermitian holomorphic vector bundle over bounded pseudoconvex domain is positive. also consider noncompact $${\mathbb {R}}^m$$ and get same result for family tube domains, which leads new method matrix-valued Prekopa’s theorem originally proved by Raufi.