So in my notes, it states that For ANY complex number $w$ in the closed ball $\overline{B(0,1)}$, there are $\alpha$ and $\beta$ such that $f(w)e^{i\alpha}=|f(w)|$ and $g(w)e^{i\beta}=|g(w)|$ where $f$ and $g$ are holomorphic on the unit closed disk.
Why is this true?
If $f(w)=0$, any $\alpha$ will work.
Otherwise, $\left|\frac{f(w)}{|f(w)|}\right|=1$ and therefore $\frac{f(w)}{|f(w)|}=e^{i\alpha}$ for some $\alpha\in\mathbb R$.