 Tangent-circle crescent conformal map

For the crescent inside $|z-2i|<2$ and outside $|z-i|\le 1$, inversion $w=1/z$ gives the horizontal strip $-1/2<\mathrm{Im}w<-1/4$. The affine map $\zeta =4w+2i$ sends it to $0<\mathrm{Im}\zeta <1$, and $e^{\pi \zeta }$ maps it bijectively onto the upper half-plane. A Cayley transform then maps it onto the unit disc. Tangency, rather than a separate enclosed hole, is why this domain is simply connected.