I'm looking for estimates of this type of sum
$$\sum_{i,j=1}^n\left|i^{-1}-j^{-1}\right|^{4k}$$
where $k\in(0,1)$.
Interesting upper/lower inequalities for
$$|i-j|^{4k}$$
could work too, but estimates of the sum would be preferable.
So far I have the classic result
$$\min\left(1,2^{4k-1}\right)\left(|i|^{4k}+|j|^{4k}\right)\leq|i-j|^{4k}\leq\max\left(1,2^{4k-1}\right)\left(|i|^{4k}+|j|^{4k}\right).$$