I was currently viewing the paper Rigidity of Holomorphic Mappings and a New Schwarz Lemma at the Boundary, by Daniel M. Burns and Steven G. Krantz. In this paper they prove the following statement.
Let $D$ denote the unit disc centered at $0\in\mathbb{C}$.
Let $\phi\colon D\rightarrow D$ be a holomorphic function from the disc to itself such that $$\phi(\zeta)=1+(\zeta-1)+O\left((\zeta-1)^4\right)$$ as } $\zeta\to1.$ Then $\phi(\zeta)\equiv\zeta$ on the disc.
They say that the exponent $4$ is strict and present a counter example of the function $$\phi(\zeta)=\zeta-\frac{1}{10}(\zeta-1)^3$$ to be a holomorphic function from the disc to itself.
So therefore arrives my question. It is clear to me that particular $\phi$ is holomorphic on $D$ and has also the "big-O" condition with the 3-exponent. However, I cannot conclude that it maps $D$ to $D$. In the paper, they state that this is due to "simple geometric arguments". Can anyone elaborate on what they mean?