Let's say I have two conformal maps $f_1, f_2$ such that $f_j:\Omega\to D_j$, where $\Omega, D_j$ are open subsets of $\Bbb{C}$. Then my question is whether there is a common technique to obtain a conformal map that maps $\Omega$ to $D_1\cap D_2$ ? My question is inspired from the following attempt:
Let $g(z) = \dfrac{1}{2}(z+\dfrac{1}{z})$ be the Joukowski map that conformally maps the complement of the open unit disk $\Bbb{D}$, $\Bbb{C}\setminus\Bbb{D}$, to $\Bbb{C}\setminus_{[-1,1]}$. Then, one can set $h(z) = ig(z)$ so that $h:\Bbb{C}\setminus\Bbb{D}\to\Bbb{C}\setminus_{[-i, i]}$. My goal is to conformally map $\Bbb{C}\setminus{\Bbb{D}}$ to $\Bbb{C}\setminus_{[-1,1]\cup[-i,i]}$.
So I am wondering if there is way to get such a map using above $h$ and $g$ ?