I'm going through the practice finals that my professor uploaded on his site, and I came across this question, and I have absolutely no one clue how to approach it and never seen anything like this on the assignments.[There are no solutions, he said he will be uploading them next week but I also have get prepared for other final exams as well.] So I'm just asking for hint to approach the question and ill make sure if my answer is correct next week. Thank you all your help.
Heres the question
Solution:
Let $n\ge3$. Note that on the contour $C$ we have $|z|=R$ and so $|f(z)|\le R^2$. Using the integral formula you are given, together with the $LM$ estimation lemma, $$|f^{(n)}(0)|\le \frac{n!}{2\pi}(2\pi R)\frac{R^2}{R^{n+1}} =\frac{n!}{R^{n-2}}\ .$$ Since $f$ is entire, this works for any radius $R$, so we can let $R\to\infty$ and then we get. . . ?