I am the author of this hint and somehow mismanaged the posting. So here it is again:

Let with small and . Then and .

Now from it follows that This shows that your inequality has the "right order of magnitude" with .

## A hint

I am the author of this hint and somehow mismanaged the posting. So here it is again:

Let with small and . Then and .

Now from it follows that This shows that your inequality has the "right order of magnitude" with .