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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.

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    $\begingroup$ 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. $\endgroup$
    – user98602
    Jan 27, 2015 at 16:20
  • $\begingroup$ Can you please explain how it is so?I am not getting it@MikeMiller $\endgroup$
    – Learnmore
    Jan 27, 2015 at 16:25
  • $\begingroup$ Is b is correct or incorrect @Learnmore how you applied Picard's theorem. $\endgroup$ Dec 13, 2018 at 7:52

2 Answers 2

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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 $|e^{z}|=1$.

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    $\begingroup$ Could you please explain to me $(c)$?How does this show that $e^z$ is bounded on imaginary axis? $\endgroup$
    – user114873
    Apr 30, 2017 at 17:24
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Part b) is incorrect. $f(z)=iz$ takes on real values on the imaginary axis.

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

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