I am having problem with the following question.
Have:
- $f(z)$ be analytic on an open set $\Omega$ with $\overline{\Delta(0, r)} \subset \Omega$.
- for zeros of $f$ in $\Delta(0, r)$ are $a_{1}, a_{2}, \ldots, a_{n}$
- $$ g(z)=\frac{r^{2}-\overline{a_{1}} z}{r\left(z-a_{1}\right)} \cdot \frac{r^{2}-\overline{a_{2}} z}{r\left(z-a_{2}\right)} \cdots \frac{r^{2}-\overline{a_{n}} z}{r\left(z-a_{n}\right)} f(z) $$
Asking for:
How to prove that the function $g(z)$ is is analytic on $\overline{\Delta(0, r)}$ except for removable singularities and does not vanish on $\Delta(0, r)$?
How to prove that $|g(z)|=|f(z)|$ for all $z$ with $|z|=r$?
All the zeros should be removable singularities, right? Then how to determine if it vanishes on $\Delta(0, r)$? or not.