[CA] Conditions at a Fixed Point for an Automorphism to be Identity

cf. Greene Ch6 Problem #17 We mentioned the condition to be identity in our previous post. Now, for a self-conformal mapping on some domain (not necessarily SCD, so Riemann Mapping Theorem fails here), when it will be identity? The answer is setting requirement on a fixed point, say P, to let its derivative be 1. Why that works? We can see its Taylor expansion at P. The 0-order and 1st-order terms simplified together to be z, so we only need to show the rest higher order terms are exactly 0. We assume there exists a non-zero higher order term, say with coefficient a_k, then check f(f(z))'s k-th power term, which has coefficient 2a_k. Inductively, f^n's k-th power term will be 2^(n-1)a_k. Because they are all self-conformal maps on the same domain (bounded region), so they share the same Cauchy estimate on the coefficient of k-th power term, which is only related to k (not n). However, 2^(n-1)a_k is increasing by the increasing of n, deriving the contradiction.