Entire function and constant

Question

Is it true that an entire function, whose $Re(s)\gt0$, then it has to be a constant? I think this is true and we could use Louville's theorem to prove this, but not sure.

Answer 1

I guess you have an entire function $f$ with $Re(f(z))>0$ for all $z$. If yes, then define $g(z)=e^{-f(z)}$.

Show that $|g(z)|<1$ for all $z$.

By Liouville, we get that $g$ is constant. Now its your turn to show that $f$ is constant.