I need a help with the following statement:
Let $D \subset \mathbb{C}^2$ be a domain containing $(0,0)$ and let $F:D \rightarrow \mathbb{C}$ be holomorphic such that $$F(z,w) = F(e^{i \theta} z,e^{i \theta} w),$$ for any $(z,w) \in D$, and $\theta \in [0,2\pi]$. Here we assume that $(e^{i \theta} z,e^{i \theta} w) \in D$, for any $(z,w) \in D$, and $\theta \in [0,2\pi]$.
Show that $F$ must be a constant.
As Martin mentioned below, I have to add the fact that $D$ contains the origin.
In one dimension the conclusion follows by the fact if a holomorphic function is constant on a path then it is constant on the whole domain. But I don't know how to verify the case in 2 dimension.