If I construct a composition of mappings that map the upper half of the unit disk conformally to the entire unit disk, then this mapping is a Riemann mapping, by the Riemann Mapping Theorem, since $D^+$ is simply connected and not the entire complex plane.
My question is: is a conformal mapping from the upper half plane to the (open) unit disk considered a Riemann mapping? What about a conformal mapping from, say, the first quadrant, to the unit disk? Naively, I'd say that these are simply connected regions that are not the entire complex plane.
My guess is that, because these pre-image regions include the point at infinity, they are somehow equivalent to being the entire complex plane - and perhaps this is more of just knowing what the Riemann sphere is? (Thus, these are not Riemann mappings.)
And, by this logic, then a mapping of a vertical or horizontal strip to the unit disk is not a Riemann mapping either.