Let $f$ be an entire function, which takes real values on the real axis and has no zeros. Suppose $f$ is bounded for $|\operatorname{Im} z| > a > 0$ where $a>0$. Is $f$ a constant?
I would be able to conclude this with Liouville's theorem if I knew that $\operatorname{Im}z$ is bounded in the strip $|\operatorname{Im} z|\le a$, but I don't see how to prove this.
I found here and in that book p.153-155 that this would be a counter-example :
$$f(z) = \int_0^\infty \frac{e^{z t}}{t^t}dt$$
which is an entire function bounded by $ \frac{1}{|c|}$ on any ray starting from the origin to infinity with an angle $\pm e^{i (\pi/2 \pm c)}$. Thus it is unbounded only on a small strip around the imaginary axis.
(I'm not 100% convinced of his proof yet)