Let $\mathbb{H}=\{z: \text{Im } z>0\}$ be the upper half plane and $0<a<1$. Find a conformal map from $\Omega_1=(\mathbb{H}\cap\mathbb{D})\backslash\{yi:y\geq a\}$ to $\Omega_2=\mathbb{H}\cap \mathbb{D}$.
By Riemann mapping theorem, there exists such a map. I've already tried functions like linear fractional transformation and $\eta=z^2$, but in vain. A hint says that one can first construct a conformal map restricted to the second quadrant and apply the Schwarz reflection theorem to get the complete map, but I still can not work it out.
Any hint or solution is highly appreciated!