Prove that a certain entire $f$ is constant

Question

Let $f$ be an entire function such that there exists $z_0, z_1$ $\mathbb{R}$-linearly independent so that $\forall z \in \mathbb{C}\ f(z+z_0)=f(z)=f(z+z_1)$. Prove that $f$ is constant.

I know that $f(z_0) = f(z_1) = f(0)$, so I guess I should prove that f is equal to $f(0)$ in a set with an accumulation point, but I don't know what to do next. Also, $f(z) = f(a z_0+b z_1)\ \forall\ a,b\ \in \mathbb{Z}$

Any hints?

Thanks, it's duplicate. I'll close it! – Guillermo Mosse May 12 '16 at 21:32 — Guillermo Mosse, May 12 '16 at 21:32

score 1 · Accepted Answer · answered May 12 '16 at 20:43

1

Prove that

$$ \sup_{z \in \mathbb{C}} |f(z)| = \sup_{t_1,t_2 \in [0,1]} |f(t_1z_1 + t_2z_2)| $$

and deduce that $f$ is bounded. Finish with Liouville's theorem.

answered May 12 '16 at 20:43

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Prove that a certain entire $f$ is constant

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