I have the following problem.
Let $f:D\to \mathbb C$ be a holomorphic function, where $D=\{z:|z|\leq 1\}.$ Let $f(z)=\sum_{n=0}^\infty c_nz^n.$ Let $l_2(A)$ denote the Lebesgue measure of a set $A\subseteq \mathbb C$ and $G=f(D).$ Prove that $l_2(G)=\pi\sum_{n=1}^\infty n|c_n|^2.$
After a long struggle I managed to come up with the following formula l_2(G)=\iint_D |f\,'(z)|^2dxdy. It looks like the right thing to use because f\,'(z)=\sum_{n=1}^\infty nc_nz^{n-1}, which is similar to what I have to prove. I understand that the $\pi$ will appear when I integrate something over the angle $\phi$ in polar coordinates. But I don't know how to find the square of the absolute value of the right-hand side to even start integrating...
EDIT: The function is supposed to be univalent (one-to-one). (I'm not perfectly sure I'm translating the term correctly. The (Polish) word in the statement of the problem was "jednolistna". I have not met it before.)