Which of the following statements are true?
a. There exits a non-constant entire function which is bounded on the upper half plane $$H=\{z\in \mathbb C:Im(z)>0\}$$
b. There exits a non-constant entire function which takes only real values on the imaginary axis
c. There exists a non-constant entire function which is bounded on the imaginary axis.
My try:
a. Let $f$ be one such function.Then $|f(z)|\leq M$ whenever $y>0$ where $z=x+iy$. I was thinking to apply Liouville's theorem but could not do it.
b. Using Picard's theorem I find such a function can't exist .
c. I can't find a counter example here.
Any help would be appreciated.
Best Answer
a. $e^{iz}$ satisfies $|e^{iz}|=e^{-\text{Im}(y)}<1$
b. $iz$ as
Tim Raczkowskitold you.c. $e^{z}$ is bounded on the imaginary axis. If $iz\in\mathbb{R}$ then $|e^{z}|=1$.