Suppose that $f(z)=\sum_{0}^{\infty} a_{k} z^{k}$ is analytic on the open unit disc $\mathbb{D}=\mathbb{D}(0,1),$ and that $f(\mathbb{D}) \subset \mathbb{D}(0,2) .($ Do not assume $f(0)=0 .)$
(a) Prove that $\sum_{0}^{\infty}\left|a_{k}\right|^{2} \leq 4 .$
(b) Using (a), show that $\left|f^{\prime}(0)\right| \leq 2,$ with equality iff $f(z)=c z,$ where $|c|=2$.
Here, Integrating $|f(z)|^{2}$ over a circle of $|z|=r,$ where $r \in(0,1)$ will leads to $\int_{|z|=r}|f(z)|^2dz=\int_{|z|=r}\sum_{0}^{\infty} a_{k} z^{k}\sum_{0}^{\infty} \bar{a_{k}} \bar{z}^{k}dz$, Now can I proceed by interchanging the summation and integral? or should I use Cauchy integral formula for derivatives at $0$?