A generalization of half-plane mappings to the ball in $\mathbb{C}^n$

Let $F$ be a normalized ($F(0)=0$, $DF(0)=I$) biholomorphic mapping of the unit ball $B \subseteq \mathbb{C}^n$ onto a convex domain $\Omega \subseteq \mathbb{C}^n$ that is the union of lines parallel to some unit vector $u \in \mathbb{C}^n$. We consider the situation in which there is one infinite singularity of $F$ on $\partial B$. 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 infinite singularity, we derive a form of the mappings in dimension $n=2$.