Entire "periodic" function

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.

Solution 1:

I'm a little wary of Liouville Theorem approaches... if you choose $f(z) = {1 \over 2}\sin(2\pi z)$ then it satisfies the conditions of the problem except $|f(z)| < e^{2\pi |z|}$ instead of $|f(z)| < e^{|z|}$.

A suggestion: try showing $f(z) = g(e^{2\pi iz})$ where $g(z)$ is analytic except at $z = 0$. Then translate the condition $|f(z)| < e^{|z|}$ into growth conditions of $|g(z)|$ as $z \rightarrow \infty$ and $z \rightarrow 0$ and show that if they occur $g(z)$ must be constant.

Solution 2:

This is a highly non-trivial theorem in complex analysis. It is called Carlson's Theorem. Roughly it states that if an entire function vanishes at integer points and have an exponential growth, then the function is zero.