I have an exercise and an approach to it's solution with a gap in it. The exercise comes from Ahlfors Complex Analysis text in Ch. 6 on Conformal mappings.
Suppose that is a Riemann map (conformal) from a simply connected domain to the open unit disk with (i.e., real and positive). Also suppose that is symmetric with respect to the real axis.
Scratch Work: For the moment assume is a mobius map and recall that mobius tranforms map Riemann circles to Riemann circles (including "lines"). Define be a reflection across the real axis and be the reflection about the circle . Form the function which maps from to . Since and is open, we can get . If then because and . So . Therefore, we have that is analytic (since conformal) and equals the identity map on . Then, analytic extension gives us that is the identity map, i.e. . So .
So there is the first gap; I don't see how to find that as claimed. If it is true, it must be that so that , but I'm not seeing it.
When we consider that is a general conformal map, there is a similar gap. In this case it is not necessarily true that the real line maps to a Riemann "circle". We have the Carathedory-Osgood extension that gives a homeomorphic extension , so the boundary of maps to the boundary of . But, again, I don't see why the segment of should map into a line (or specifically into the real axis) in the disk so that we can conclude that the reflection is in fact .
Thanks in advance for your consideration!