Existence of a non-constant entire function

Question

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.

Answer 1

a. $e^{iz}$ satisfies $|e^{iz}|=e^{-\text{Im}(y)}<1$

b. $iz$ as Tim Raczkowski told you.

c. $e^{z}$ is bounded on the imaginary axis. If $iz\in\mathbb{R}$ then $|e^{z}|=1$.

Answer 2

Part b) is incorrect. $f(z)=iz$ takes on real values on the imaginary axis.