If $f^3,~\bar{f}^2$ are holomorphic on a domain $D$, then $f$ is constant on $D$.

Question

Let $D$ be an open and connected region on the complex plane and $f:D\rightarrow \mathbb{C}$ be a function such that $f^3,~\bar{f}^2$ are holomorphic. Prove that $f$ is constant on $D$.

Attempt. A classic technique is such cases is to show that $f'(z)=0$ on $D$. I have thought of $Ref$ or $Im f$, in case they are constant, but I don't seem to get the desired result that way.

Thanks in advance.

Answer 1

Hint: If $f^3$ and $\overline{f}^2$ are holomorphic, so is $|f|^{12}$.