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Let $f$ be an entire function. For which of the following cases $f$ is not necessarily a constant

  1. $\operatorname{im}(f'(z))>0$ for all $z$
  2. $f'(0)=0$ and $|f'(z)|\leq3$ for all $z$
  3. $f(n)=3$ for all integer $n$
  4. $f(z) =i$ 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

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In your statement number $4$: what's $n$? – Kevin Carlson Oct 27 '12 at 19:19
up vote 3 down vote accepted

You're right for $1$, where the $c$ you mention should have positive imaginary part. For example $f(z)=iz$ does the job.

For $2$, $f$ need not be identically $0$. $f(z)=2$ satisfies the requirements, for instance. $f$ does have to be constant, though, because if $f$ is entire, $f'$ is too, and so by Liouville's theorem since $f'$ is bounded it's constant, and the other condition forces it to be $0$ everywhere.

You're right on $3$.

For $4$, I'm still not sure what $n$ is. My guess is that this will be an identity theorem application, and that $f$ will be $i$ on a set with an accumulation point, which will force it to be $i$ everywhere.

But as it's written now, for $n$ is a fixed integer you could get $f$ non-constant in the same way as you did in number $3$: set $$f(z)=i\cos(2n\pi(z-1))$$

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