Couple of questions.
I need to show that if f: D -> D is holomorphic has two distinct fixed points, the f(z)=z. (D is a unit disk)
Now, can we assume that one of these two fixed points is an origin?
Because then the result follows from Schwarz lemma.
Also, to answer the question "Must every holomorphic function f: D -> D have a fixed point?", would it be correct to consider a holomorphic map from H to H which doesn't fix any point (for example g(z) = epsilon * z, where epsilon >0) and then consider a map F(g(G(z))) where F and G are canonical conformal maps from D to H and H to D?
Since F and G are one-to-one this should provide an example of a hol. function that doesn't have fixed points. .