I want to prove that given $f \in O(D_{R})$ (holomorphic in a disk of radius $R$ centered in 0), then $D_{\frac{R|f'(0)|^2}{4 ||f'||_{\infty}}} \subset f(D_{R})$. Now, following the demonstration, you take $c \notin f(D_R)$ and define $\psi$ an holomorphic function such that $\psi(z)^2 = 1- f(z)/c$ on $D_R$. Now, given $z=re^{i \theta} \in D_R$ I write $\psi(z)$ as:
$$\psi(r e^{i \theta})= \sum_{n=0}^{\infty} \frac{\psi^{(n)}(0)}{n!}r^n e^{in \theta}$$
Then I want to do two things with that represetation: integrate from $0$ to $2 \pi$ with respect to $\theta$ (exchanging integral and series) and multiply that series with the series representing $\bar \psi(z)$ to obtain a series representation of $|\psi(z)|^2$.
The question is: how do I prove I can do that? I require uniform convergence to do the former and absolute convergence to do the latter, if I am not mistaken.