The range of holomorphic maps at boundary points

We prove a boundary version of the open mapping theorem for holomorphic maps between strongly pseudoconvex domains. That is, we prove that the local image of a holomorphic map \(f{:\,}D\rightarrow D'\) close to a boundary regular contact point \(p\in \partial D\) where the Jacobian is bounded away from zero along normal non-tangential directions has to eventually contain every cone (and more generally every region which is Kobayashi asymptotic to a cone) with vertex at \(f(p)\) .