Univalent C 1 Mappings on the Unit Ball in CR

Let B be the unit ball with respect to an arbitrary norm on C ⁿ . We give analytic sufficient conditions for a local diffeomorphism of C ¹ class on B to be univalent and to have a Φ-like image. When Φ is holomorphic and f is a locally biholomorphic mapping with f(0) = 0, Df(0) = I, the conditions are also necessary conditions. We also give analytic sufficient conditions to be univalent and to have an Archimedean spirallike image or a hyperbolic spirallike image.