Cartan's uniqueness theorem

Remarks

The proof uses iteration. Take $p = 0$ and suppose that $f(z) = z + P_k(z) + \cdots$, where $P_k$ is the first nonzero homogeneous term of degree $k \ge 2$ in the Taylor expansion of $f - \id$. The $m$-th iterate of $f$ then has the expansion $z + m P_k(z) + \cdots$. All iterates map $D$ into the bounded set $D$, so the Cauchy estimates bound their Taylor coefficients independently of $m$, which forces $P_k = 0$.

A consequence: if $g, h$ are biholomorphisms of $D$ onto a domain $D'$ with $g(p) = h(p)$ and the same Jacobian matrix at $p$, then $g = h$. So an automorphism of a bounded domain is determined by its value and its derivative at one point.

Boundedness is necessary. On $\C^2$ the map $f(z_1, z_2) = (z_1, z_2 + z_1^2)$ fixes $0$ and has the identity as Jacobian matrix at $0$, but it is not the identity map.

Going deeper

For a student

The theorem is the basic tool for studying the automorphism group of a bounded domain. It shows that the map sending an automorphism fixing $p$ to its Jacobian matrix at $p$ is injective. With it one proves that automorphisms of a bounded circular domain fixing the origin are linear maps. It is also a step toward showing that the automorphism group of a bounded domain is a Lie group.