Unbounded convex mappings of the ball in $\mathbb{C} ^n$

In this paper, we study univalent holomorphic mappings of the unit ball in $\mathbb{C} ^n$ that have the property that the image $F(B)$contains a line $\{tu: t \in \mathbb{R}\}$ for some $u \in \mathbb{C} ^n$, $u \neq 0$. We show that under certain rather reasonable conditions, up to composition with a holomorphic automorphism of the ball, the mapping $F$is an extension of the strip mapping in the plane to higher dimensions.