||Let p : X. Y be a surjective holomorphic mapping between Kahler manifolds. Let D be a smoothly bounded domain in X such that every generic fiber Dy := Dn p-1(y) for y. Y is a strongly pseudoconvex domain in Xy := p -1(y), which admits the complete Kahler-Einstein metric. This family of Kahler-Einstein metrics induces a smooth (1, 1)-form. on D. In this paper, we prove that. is positive-definite on D if D is strongly pseudoconvex. We also discuss the extension of. as a positive current across singular fibers.