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I'm studying for a qualifying exam in complex analysis and this question was asked in an old exam: Let $f$ be an entire function such that $|f(z)| \leq |\sqrt{z}|$ for sufficiently large $z$. Evaluate $f(2017)$. Justify your answer.

So far I can show that $f$ must be constant (Cauchy's estimates). However I don't see a way to evaluate $f(2017)$. If the assumption was $|f(z)| \leq |\sqrt{z}|$ for all $z$, then we would have $f(2017)=0$. However I don't see how to proceed with the question as stated. Am I missing something or is the question formulated incorrectly?

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    $\begingroup$ This seems weird. $|\sqrt{z}|=|\sqrt{re^{i\theta}}|=\sqrt{r}$ which goes to infinity as $z$ moves further from the origin. As such, any constant function would satisfy the condition $|f(z)|\leq |\sqrt{z}|$ for sufficiently large $z$. $\endgroup$
    – QC_QAOA
    Jan 1, 2020 at 3:53
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    $\begingroup$ It's not unheard of for a professor to write an erroneous qual question. I personally took one in which almost half the test questions had to be removed after the exam date because there was no way to save them. In this case, I would give the proof that $f(z)$ must be constant and then show that $f(z) = C$ satisfies the assumptions for any $C \in \mathbb{C},$ and then comment $f'(2017) = 0$ while $f(2017)$ is undetermined. $\endgroup$ Jan 1, 2020 at 4:01

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The question is formulated incorrectly. If you take $f(z)=c$ for some constant $c$, as you noted, $|f(z)|\leq |\sqrt z|$ for all $z$ with $|z|\geq c^2$, so all such functions work. It seems you have also proven that these are the only such functions. Noting such things is essentially all you can do for such a problem, and would hopefully give you full points from any reasonable grader.

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