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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

$$\min\left(1,2^{4k-1}\right)\left(|i|^{4k}+|j|^{4k}\right)\leq|i-j|^{4k}\leq\max\left(1,2^{4k-1}\right)\left(|i|^{4k}+|j|^{4k}\right).$$

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  • $\begingroup$ Restrict to i < j so you can get rid of the absolute values. $\endgroup$
    – marty cohen
    Commented Oct 10, 2019 at 15:37
  • $\begingroup$ What would it change in the inequality? I would still need a difference and thus absolute values. Or are you suggesting other approach? $\endgroup$
    – TheGrandDuke
    Commented Oct 10, 2019 at 16:35

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