bounded holomorphic function on right-hand half plane with periodic 1 must be a constant function Question: Suppose $f$ is a bounded holomorphic function on $\{z\in\mathbb{C}:\text{Re}(z)>0\}$, and $f$ is periodic with period 1, that is $f(z+1)=f(z)$ when $\text{Re}(z)>0$. Prove that $f$ must be a constant function.
Can we extend $f$ to be an entire function on the whole complex plane? For example, can we get an entire function by using even extension? If that is possible, then the statement can be proved by the Liouville theorem. 

For any $z \in \mathbb{C}$, let $n_z$ be the smallest integer such that $\Re(z+n_z) > 0$. Define $F(z) = f(z+n_z)$. This is an analytic continuation of $f$ to the entire complex plane. By Liouville, $F$ must be constant, and hence, $f$ was constant to begin with.
