Find a conformal (biholomorphic) map between the unit disc $D = \{z : |z|<1\}$ and the domain $U = \left\{z \in \mathbb{C} : |z| < 1 \mbox{ and } \operatorname{ Im}(z) > \frac{1}{\sqrt{2}}\right\}$.
The first thing I thought was to consider $f_1(z) = z+i/\sqrt{2}$ applied to $U$ (so shifting $U$ so it's contained in the upper-half-plane), and denoting $U_1 = f_1(U)$. Then consider the maps $1/z$ and $\log(z)$ on $U_1$. But $1/z$ didn't result in a space that looked familiar. $\log(z)$ maps $U_1$ to something similar to a horizontal half-strip, but not close enough that I could see what to do.
Edited for typos and clarity.