Home » Articles posted by John Khan

# Author Archives: John Khan

## Homogeneity of inequality

Let . Prove that:

Solution

Due to homogeneity we may assume . Thus there exist positive such that

Hence,

## Galois theory … of the Euler’s totient function

Let and let be an -th primitive root of unity. Prove that

where denotes the Euler’s totient function.

Solution

We have

where since is the minimal polynomial of over .

The result follows.

## Complex inequality

Prove that forall and it holds that

Solution

Well,

Done!

## Bound of derivative

Let . Let denote the -th derivative. Prove that

Solution

We note that

Hence,

## Searching for the …function

Find all functions such that and

Solution

First of all we note that

Thus,

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.