This is part of an assignment so I know it's definitely got an answer, but I can't for the life of me figure out what the question is asking.
I may be interpreting this wrong, but to me the original domain is the unit disc, except it excludes the negative imaginary axis. I think this defines that the domain is bounded by a curve that approaches the negative imaginary axis from certain directions, but I don't know how to map what is basically a circle minus one infinitesimally small point...onto a full circle.
I'm new to complex analysis (and I wasn't that great at maths to start with) so sorry if this is a trivial issue, but the only other notes or examples I've seen are mapping segments / quadrants onto unit circles, or well defined domains, not mapping a domain onto effectively itself with a very minor technical tweak.
$D \setminus (0, -i)$ is not an open set, I assume the question is about $D \setminus [0, -i)$.
Take $\sqrt z$ to be analytic on $\mathbb C \setminus [0, -i \infty)$ and apply $z \mapsto \sqrt z$ to $D \setminus [0, -i)$. Then consider that a circular segment can be mapped to a sector bounded by two rays by a Mobius transformation (if one of the corner points is mapped to $\infty$), which reduces the problem to one of the cases that you mentioned.