A generalization of half-plane mappings to the ball in Cn

Let F be a normalized (F(0) = 0, DF(0) = I) biholomorphic mapping of the unit ball B subset of C-n onto a convex domain.. Cn that is the union of lines parallel to some unit vector u is an element of C-n. We consider the situation in which there is one infinite singularity of F on. B. In one case with a simple change-of-variables, we classify all convex mappings of B that are half-plane mappings in the first coordinate. In the more complicated case, when u is not in the span of the in finite singularity, we derive a form of the mappings in dimension n = 2.