TITLE: Entire "periodic" function
QUESTION [13 upvotes]: I am studing for exams and am stuck on this problem.  
 
Suppose $f$ is an entire function s.t. $f(z) =f(z+1)$ and $|f(z)| < e^{|z|}$.
Show $f$ is constant.

I've deduced so far that:
a) $f$ is bounded on every horizontal strip 
b) for every bounded horizontal strip of length greater than 1 a maximum modulus must occur on a horizontal boundary.

REPLY [2 votes]: Consider 
$$
g(z) = \frac{f(z) - f(0)}{\sin(\pi z)}
$$
This is an entire function, since $\sin(\pi z)$ has poles at the integers which are cancelled by the zeros of $f(z) - f(0)$ which also occur at every integer. We have $g(z + 2\ell) = g(z)$ and also $g(x + iy) \rightarrow 0$ when $|y| \rightarrow \infty$ and $|x| \leq B$ for any fixed $B$. Therefore $g$ is bounded. Hence by Liouville $g = C$ with $C$ constant. We must have $C = 0$ because otherwise $f$ is of order greater than $e^{|z|}$ (i.e it would be of order at least $e^{|\pi z|}$). Therefore $C  = 0$ and $f(z) = f(0)$ as desired.