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?
Best Answer
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.