# Does there exists non-constant entire function with the following conditions?

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?

• What are you not sure about ? – reuns Jul 27 '17 at 7:15
• If you want to be sure, explain what you are not sure about.. – reuns Jul 27 '17 at 7:19
• I, for one, see nothing wrong with anything you've done here. – Theo Bendit Jul 27 '17 at 7:29
• Yes. It's correct. My choice for 2. would be $f(z)=3\cos^2 \pi z.$ But there are infinitely many examples. – DanielWainfleet Jul 27 '17 at 7:55
• 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. – DanielWainfleet Jul 27 '17 at 8:01