[Math] an analytic function from unit disk to unit disk with two fixed point

Question

prove that if $f:\mathbb{D}\rightarrow\mathbb{D}$ is analytic with two distinct fixed point then $f$ is identity.

I thought if one of the fixed points was zero by schwarz lemma this statement is easily proved.

but what can I do if fixed points were nonzero?

Answer 1

I thought if one of the fixed points were zero by the Schwarz lemma this statement is easily proved.

Good. So it remains to reduce the general case to the known special case.

Let $T$ be an automorphism of $\mathbb{D}$. What do you know about $T^{-1}\circ f \circ T$? Can you find a condition on $T$ that reduces the problem to the known case?