Existence of a non-constant entire function

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.

• All three of these are possible. If it said "real axis" instead of imaginary axis for the last two, could you do it? For the first, pick a conformal equivalence with the disc. – user98602 Jan 27 '15 at 16:20
• Can you please explain how it is so?I am not getting it@MikeMiller – Learnmore Jan 27 '15 at 16:25
• Is b is correct or incorrect @Learnmore how you applied Picard's theorem. – Pranita Gupta Dec 13 '18 at 7:52

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 $|e^{z}|=1$.
• Could you please explain to me $(c)$?How does this show that $e^z$ is bounded on imaginary axis? – user114873 Apr 30 '17 at 17:24
Part b) is incorrect. $f(z)=iz$ takes on real values on the imaginary axis.