Holomorphic injection between open ball and the plane minus an interval Let $\mathbb C$ be the complex numebrs. I am looking for a holomorphic map that bijectively maps the open ball
$$
\{ z \in \mathbb C \mid \lvert z \rvert < 1\}
$$
To the set
$$
\mathbb{C} / [-1,1].
$$
I have tried with the sine defined on a rectangle-like region $R$ of the plane to later compose it with some biyection from $R$ to the open ball, but at some point I need some kind of logarithm defined on the open ball. The original problem only required $f$ to be injective and that can be more or less solved with a sine function. It is even possible? Have in mind that I don't have strong theorems like the Riemann mapping theorem or anything related.

This is not possible. If what you want existed we’d have a biholomorphic map from the disc to $\mathbb{C}\setminus [-1,1]$. The simple-connectedness assumption of the Riemann mapping theorem turns out to be necessary and sufficient.
Suppose such an $f$ existed. Take a simple closed curve $\gamma$ in $\mathbb{C}/[-1,1]$ with winding number 1 around zero. Pulling back this curve through $f$ gives a curve $\gamma^*$ in $\mathbb{C}$. Applying the argument principle to $f$ on the curve $\gamma^*$ shows that $f$ must have a zero on the interior of $\gamma^*$. This is a contradiction as zero is not in the range of $f$.
