Exercise for the identity theorem

Question

Question : If $f$ and $g$ are analytic functions on domain $\mathbb{D}$ and so is the product $\overline{f}g$ on $\mathbb{D}$. Then, $f$ is a constant or $g\equiv 0$ identically

I do not know how to start. If not an answer, I hope the hint for Question.

1. $\overline{f}$ means a conjugate function for $f$

2. This exercise is included in the section for The identity Theorem.

Answer 1

Hint: If $g\not \equiv 0,$ then there is an open disc $U\subset D$ where $g$ is nonzero. On $U$ we can divide by $g$ to see that $\overline f$ is analytic on $U.$