How to deal with $|f(z)|^2$ under integral
A related problem. Using the fact that $|z|^2=z\bar{z}$, we have
$$ \frac{1}{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^2d\theta=\frac{1}{2\pi}\int_0^{2\pi}f(re^{i\theta}) \overline{f(re^{i\theta})} d\theta$$
$$=\frac{1}{2\pi}\int_0^{2\pi}\sum_{n=0}^{\infty}c_nr^n e^{in\theta} \overline{\sum_{m=0}^{\infty} c_mr^m e^{im\theta}} d\theta $$
$$=\frac{1}{2\pi} \sum_{n=0}^{\infty}c_nr^n \sum_{m=0}^{\infty}\bar{c_m}r^m\int_{0}^{2\pi}e^{i(n-m)\theta} d \theta $$
The other fact you want to use is $\int_{0}^{2\pi}e^{in\theta}e^{-im\theta}=2\pi$ for $n= m$ and $0$ otherwise.