A conformal mapping onto a region bounded by convex contours (Ahlfors)

Let $\gamma_k$ be the image of $C_k$ under $p+q$.

Since $\gamma_k$ is convex, there are only three types of points on the $w$-plane.

• On-curve: $w$ is on $\gamma_k$.
• External: We can draw a straight line connecting $w$ and infinity without intersecting $\gamma_k$. Obviously the winding number of $\gamma_k$ with respect to an external point is 0.
• Internal: We cannot draw a straight line connecting $w$ and infinity without intersecting $\gamma_k$.

Now let us assume that $w$ is a value of $p+q$, i.e. $p(z) + q(z) = w$ for a certain $z \in \Omega$.

$w$ cannot be internal, because this would be in contradiction with the fact that $\Omega$ is connected and $z_0 \in \Omega$ is mapped to infinity.

If $w$ is external, we are done.

If $w$ is on-curve, either it is on a slit, or it has internal points in its neighborhood.

In the first case, we can make the same argument under (17) on Page 260 in Ahlfors that the principle part of the following integral must vanish.

$$\int_{C_k} \frac{p'(z) + q'(z)}{p(z)+q(z)-w} dz$$

The second case is impossible because of the following corollary (Ahlfors, Page 132).

Corollary 1. A nonconstant analytic function maps open sets onto open sets.