Suppose that U is a simply-connected open domain inCand assume

that $\displaystyle f,g : U \rightarrow U$ 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 $\displaystyle f(z_i) = g(z_i)$ for i = 1, 2 and $\displaystyle z_1 \neq z_2$ 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.