Find all entire functions with real part greater than 1 and imaginary part less than -1

Question

I'm stuck on the following question:

Find all entire functions $f$ such that $Re(f) >1$ and $Im(f)<-1$.

Unfortunately this doesn't seem like a problem that can be quickly solved by liouville's theorem, so the next idea I had was to use the power series expansion of $f$ and try to figure out something with that, but I'm not sure how to proceed since using the power series is a little messy to do if you have to consider the real and imaginary parts separately.

source: Spring 1992

Answer 1

Note that$$\left|e^{-f(z)}\right|=e^{-\operatorname{Re}f(z)}<e^{-1}.$$Therefore, $e^{-f}$ is bounded and so…