Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
    
In your statement number $4$: what's $n$? –  Kevin Carlson Oct 27 '12 at 19:19
add comment

1 Answer 1

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 $$f(z)=i\cos(2n\pi(z-1))$$

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.