Is it true that if $\frac{u\circ\Phi}{u}$ is holomorphic for any automorphism $\Phi$, then $u$ is holomorphic?

Question

Let $G$ be a homogeneous (or even symmetric) domain in $\mathbb{C}^{n}$ and $u:G\to \mathbb{C}\backslash\{0\}$ is such that $\frac{u\circ\Phi}{u}$ is holomorphic for any biholomorphism $\Phi:G\to G$. Is it true that then $u$ is holomorphic?