cf. an inequality involving sigma and tau.

Let be a positive integer. By the weighted AM-GM inequality, one has

,

where all sums and products are taken over the positive divisors of . This means

.

Considering the analytic behaviour of the function one deduces

for any , with equality iff is prime or or . (By convention we take an empty product to be .) Combining this with the first inequality in this post we obtain

,

i.e.,

.

This strengthens the first boxed inequality from this post.

### Like this:

Like Loading...

*Related*