I am looking for a bijective holomorphic function from the upper half plane to $H_1=\{z \text{ }|\text{ } 0<\text{Im}(z)<1\}$.
Context:
Let $D$ be the unit open ball and $D_{1/2}(-1/2)$ be the open ball of radius $1/2$ around $z=-1/2$. I am trying to find a bijective holomorphic function from $D$ to $D-\overline{D_{1/2}(-1/2)}$.
My idea:
(1) The function $$f:z\mapsto \frac{i-z}{i+z}$$ is a bijective holomorphic function between the upper half plane $H$ and D.
(2) The function $f$ maps the the horizontal line $\text{Im}(z)=C$ to the circle of radius $\frac{1}{C+1}$ centered at $(-\frac{C}{C+1}, 0)$.
(3) From (2) it follows that $H_2=\{z \text{ }|\text{ } \text{Im}(z)\geq 1\}$ is mapped to $\overline{D_{1/2}(-1/2)}$ under $f$. Therefore $H\setminus H_2$ is mapped to $D-\overline{D_{1/2}(-1/2)}$. But $H\setminus H_2=H_1$. If we can find a bijective holomorphic function between $H$ and $H_1$, we are done.
How to find such a function?
Edit:
If we can find a bijective holomorphic function $g$ from $D$ to $H_1$, then the function $f\circ g$ will be a function from $D$ to $D-\overline{D_{1/2}(-1/2)}$.
The existence of such a mapping is guaranteed by the Riemann mapping theorem, since both the upper halfplane and the strip $H_1$ are simply connected and not equal to $\Bbb C$.
For a concrete mapping, start with an analytic branch of the logarithm in the upper halfplane, e.g.
$$ \log z = \log |z| + i \arg z \text{ with } 0 < \arg z < \pi \, . $$