Prove that $A=\pi\sum\limits_{n=0}^{\infty}{n|a_n|^2}$

Question

Let $f(z)=\sum\limits_{n=0}^{\infty}{a_nz^n}$ is an univalent analytic function on the open disk $\Delta(0,1)=\{z:|z|<1\}$, such that $G= f(\Delta(0,1))$. If $A$ is the area of the region $G$, prove that:

$$A=\pi\sum\limits_{n=0}^{\infty}{n|a_n|^2}$$

I found this question from the book "Fundamentals of Complex Analysis", written by Liao Liangwen. I have no idea about this. So I want to get some help.

Thanks.