Let $f:\mathbb{\Omega}\to\mathbb{\Omega}$ be an analytic function such that there exist $z_0$ and $z_1$ $\mathbb{R}$-linearly independent$f(z+z_0) = f(z)$ and $f(z+z_1) = f(z)$ for all $z$. Show that $f$ is constant.
I can show that $f(nz_0+mz_1) = f(0)$ for $n$ and $m$ natural numbers, and I know that if I could show it for $n$ and $m$ rational, it will be enough to conclude. But I'm not really sure how to continue or whether that's the right way or not (maybe I could use the maximum modulus principle o some similar theorem). Any help will be appreciated
