The mapping $F:z\mapsto e^{i\pi z}$ maps $S\colon=[0 < \Re z < 1]$ conformaly onto $H^+$. Find for any $w\in S$ a conformal mapping $R_w:S\rar B^2$, which is onto and $R_w(w)=0$.
Put $z=x+iy$, then $\Im(e^{i\pi z})=\Im(e^{i\pi x-\pi y})=e^{-\pi y}\sin(\pi x) > 0$.
Conversely for any $u\in H^+$ the equation $e^{i\pi z}=u$ is solved by
$$
z=-i\pi^{-1}\log u
=\pi^{-1}(-i\log|u|+\arg(u))~.
$$
Since $\Im u > 0$: $\arg(u)\in(0,\pi)$ and therefore:
$$
\Re(z)=\Re(-i\log|u|/\pi+\arg(u)/\pi)=\arg(u)/\pi\in(0,1),
\quad\mbox{i.e.}\quad
z\in S~.
$$
2. For any $w\in S$ a suitable mapping $R_w:S\rar D$ is given by
$$
R_w(z)=\frac{e^{i\pi z}-e^{i\pi w}}{e^{i\pi z}-e^{-i\pi w}}~.
$$