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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.

1 Answers 1

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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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    Be careful with your absolute value signs! You know $z^2f(z)-3+e^z$ is entire, not its absolute value.2012-08-26