Thread: Complex Analysis -- functions on a connected set

Suppose that U is a simply-connected open domain in C and assume
that are one-to-one and onto maps which are holomorphic mappings with the property that f' and g' are non-zero for all points of
U. Prove that if for i = 1, 2 and then f = g.

I am not sure where to start on this one. I know that if f and g are equal on a set which is not discrete then they are the same, but in this case the set is just two elements. I'm not sure what piece of the puzzle I am missing -- any help would be appreciated.

Let be the upper half-space. We know that automorphisms of are of the form



then (working with ) assume is such a function with two fixed points. Without loss of generality assume then . Let be the other fixed point then



and this leads to, assuming , or so we have or .

Now, since automorphisms of are affine transformations the result in this case is obvious.

Conclude by the Riemann mapping theorem.