Let $z_0$ be in an open set $U \subseteq \mathbb{C}$. Let $ f, \phi: U \rightarrow \Delta(0,1)$. Suppose $\phi$ is bijection and $$\phi(z_0)=0,\ \phi'(z_0) \neq 0.$$ Assume, $f(z_0)=0$, prove that $$|f'(z_0)| \leq |\phi'(z_0)|,$$ with the equality if and only if $$f(z)= \lambda \phi(z)$$ with $|\lambda|=1$.
Here is where I am struggeling: We have here $\phi^{-1} $ is analytic. My ideas do not work with having this property, I am trying to use Schwarz's lemma but still have some problems. I am thankful for any help with that.