# 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, 2015 at 16:20
• Can you please explain how it is so?I am not getting it@MikeMiller Jan 27, 2015 at 16:25
• Is b is correct or incorrect @Learnmore how you applied Picard's theorem. Dec 13, 2018 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? Apr 30, 2017 at 17:24

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

• Can you please help me with a and c Jan 27, 2015 at 16:29
• Why do you say part (b) is incorrect? It is correct right? The statement is true. Jul 26, 2017 at 20:17