Show that any conformal self-map of the upper half-plane has the form

, ,

where are real numbers satisfying . When do two such coefficient choices for determine the same conformal self-map of the upper half-plane?

In this section we have covered Pick's Lemma and we have covered that fractional linear transformations have the form . However, I do not see how conformal self-maps in the upper half-plane have this form too. I need some hints on how to start this one. Thanks in advance.