Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have been struggling to find a solution for this problem:

Find all the entire analytic functions $f(z)$ (analytic in the complex plane) that satisfy the condition $|z^2f(z)-3+e^z|\leq3$ for all $z \in \mathbb{C}$.

Any ideas?

Thank you in advance.

share|cite|improve this question
up vote 5 down vote accepted

If $f(z)$ is entire, then $g(z)=z^2f(z)-3+e^z$ is entire. But $g$ is bounded and entire, so by Liouville's theorem it reduces to a constant. Solving for $f(z)$, we find it has to have a pole at $0$, so there are no solutions.

share|cite|improve this answer
to make sure i'm getting it right, Liouville's theorem says that a bounded entire function is a constant, so $|z^2f(z)-3+e^z|$ is a constant since the expression is entire, therefore solving $z^2f(z)-3+e^z=c$ well yield $f(z)=(1/(z^3))(c+3-e^z)$ but then $f(z)$ has a pole at $z=0$, which is contradictory to the given info that it is entire, therefore there is no solutions for the problem. that sounds right, thank you very much. – Jar Aug 26 '12 at 1:47
Be careful with your absolute value signs! You know $z^2f(z)-3+e^z$ is entire, not its absolute value. – Potato Aug 26 '12 at 2:32

Your Answer


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.