Show that any biholomorphic map from $U = \{z \in \mathbb{C} : Im(z)>0\}$ to $U$ is a Mobius transformation.
I know this statement to be true, just having a hard time directly proving this, I think I am confusing myself on how to start. Can someone give me a hint on what to do here or show me how I would go about this?
My fist thought was to characterize all biholomorphic maps from $U$ to $U$ and hope that I see the form of a mobius transformation fall out, from there I would show that the determinant of the coefficients was equal to one, but I wasn't sure how to characterize these maps and even if I could I think I would have a hard time directly showing the determinant of the coefficients was one. I am a first time complex analysis student and my professor has decided to go out of order and kind of out of our textbook to teach conformal mappings and mobius transforms so I am a little confused here, any help is greatly appreciated.