Let $D(a, R)$ denote the disc $|z-a|<R$ and let $f: D(a, R) \rightarrow \mathbb{C}$ be a holomorphic function. Using Cauchy's integral formula show that for every $r \in(0, R)$

$$f(a)=\int_{0}^{1} f\left(a+r e^{2 \pi i t}\right) d t$$

Deduce that if for every $z \in D(a, R),|f(z)| \leqslant|f(a)|$, then $f$ is constant.

Let $f: D(0,1) \rightarrow D(0,1)$ be holomorphic with $f(0)=0$. Show that $|f(z)| \leqslant|z|$ for all $z \in D(0,1)$. Moreover, show that if $|f(w)|=|w|$ for some $w \neq 0$, then there exists $\lambda$ with $|\lambda|=1$ such that $f(z)=\lambda z$ for all $z \in D(0,1)$.