# Let $f$ be an entire function such that $|f(z)| \leq |\sqrt{z}|$ for sufficiently large $z$. Evaluate $f(2017)$.

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?

• 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 $z$. Jan 1, 2020 at 3:53
• 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. Jan 1, 2020 at 4:01

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.