Let . Prove that:
Due to homogeneity we may assume . Thus there exist positive such that
Let and let be an -th primitive root of unity. Prove that
where denotes the Euler’s totient function.
where since is the minimal polynomial of over .
The result follows.
Prove that forall and it holds that
Let . Let denote the -th derivative. Prove that
We note that
Find all functions such that and
First of all we note that
Since the integrand is positive it only remains that
Setting and at the last equation we have that:
Since is positive we conclude that
which satisfies the given conditions.