Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be an analytic function. Pick out the statements for which $f$ is not necessarily constant.

  1. $\operatorname{Im}(f'(z))>0$ for all $z\in \mathbb{C}$

  2. $f(n)=3$ for all $n\in \mathbb{Z}$

  3. $f'(0)=0$ and $|f'(z)|\leq 3$ for all $z\in \mathbb{C}$

My attempt: For (1) $f(z)=iz$ is a function which satisfies (1) but is not constant.

For (2) $f(z)=3+\sin(n\pi z)$ is also a non-constant function satisfying (2)

For (3), $f$ is entire implies $f'$ is entire. It is given that $f'$ is bounded hence $f'$ is constant but then $f'(0)=0$. Therefore, $f'=0$ and therefore $f$ is constant.

- Are my attempts correct?

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    $\begingroup$ What are you not sure about ? $\endgroup$ – reuns Jul 27 '17 at 7:15
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    $\begingroup$ If you want to be sure, explain what you are not sure about.. $\endgroup$ – reuns Jul 27 '17 at 7:19
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    $\begingroup$ I, for one, see nothing wrong with anything you've done here. $\endgroup$ – Theo Bendit Jul 27 '17 at 7:29
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    $\begingroup$ Yes. It's correct. My choice for 2. would be $f(z)=3\cos^2 \pi z.$ But there are infinitely many examples. $\endgroup$ – DanielWainfleet Jul 27 '17 at 7:55
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    $\begingroup$ BTW. Regarding 1., the "little Picard theorem" is that if $g:\mathbb C\to \mathbb C$ is analytic and not constant then the image of $ f $ is $\mathbb C$ \ $S$ where $S$ has at most $1$ member. Applying this with $g=f',$ if $Im (f'(z))>0$ for all $z$ then $f'$ is constant so $f$ is linear. $\endgroup$ – DanielWainfleet Jul 27 '17 at 8:01

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