Prove that an entire function with $Imf\le (Ref)^2$is constant

Question

This is an old question from Ph.D Qualifying Exam of Complex Analysis.

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Without using Picard's theorem directly, prove that if $f$ is an entire function such that $\text{Im}f(z)\le (\text{Re}f(z))^2$ for all $z\in \mathbb{C}$ then $f$ is constant.

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My attempt: I'd like to apply Liouville's theorem, but I can't find the entire function to apply the theorem. First I tried for $e^{f^2}$, but its real part becomes $e^{(Ref)^2-(Imf)^2}\ge e^{Imf-(Imf)^2}$ so I failed to find the bound.

How should I do?

Thanks in advance!

Answer 1

The image of $f$ does not intersect a disc centred at $i$. Therefore $z\mapsto 1/(f(z)-i)$ is entire and bounded. Use Liouville.