Let $f$ be an entire function. For which of the following cases $f$ is not necessarily a constant
- $\operatorname{im}(f'(z))>0$ for all $z$
- $f'(0)=0$ and $|f'(z)|\leq3$ for all $z$
- $f(n)=3$ for all integer $n$
- $f(z) =i$ when $z=(1+\frac{k}{n})$ for every positive integer $k$
I think 1 is true since $f'(z)=$constant so $f(z)=cz$ for some $c$
For 2, $f=0$
For 3, $f$ is not constant since $f(z)=3 \cos2\pi z$
I have no idea for 4
am i right for other three options