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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.

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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.

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  • $\begingroup$ 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. $\endgroup$
    – Jar
    Aug 26 '12 at 1:47
  • $\begingroup$ Be careful with your absolute value signs! You know $z^2f(z)-3+e^z$ is entire, not its absolute value. $\endgroup$
    – Potato
    Aug 26 '12 at 2:32

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