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Given $a_1,\ldots,a_n,x_1,\ldots,x_n\in\mathbf{R}$ with $\sum_{i=1}^{n}a_i=0$, prove $\sum_{i,j}a_ia_j\ln|x_i+x_j|\leq 0$.

This is a thought of another question: $\sum_{i,j}a_ia_j|x_i-x_j|^{p}\leq 0$, which holds when $0<p<2$. It's easy to get by: $\frac{\pi}{2\Gamma(1+s)\sin(\pi s/2)}|a|^s=\int_{0}^{\infty}\frac{1-\cos at}{t^{1+s}}dt$ but I get trouble when consider $\ln |x_i+x_j|$ instead of $|x_i-x_j|^{p}$, and it seems difficult to do with $\sum_{i,j}a_ia_j\ln\int_{0}^\infty\frac{2}{\pi}(\frac{1-\cos(x_i+x_j)t}{t^2})dt$

metamorphy
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Justin
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  • Hi and welcome to the site! It is advisable that you take a tour to see what we are about. Since this is a site that encourages and helps with learning, it is best if you show your own ideas and efforts in solving the question. Can you edit your question to add your thoughts and ideas about it? Don't worry if it's wrong - that's what we're here for. Here's a quick guide (if nothing else, read up the part on "avoiding no-clue questions"). – 5xum May 16 '22 at 13:36
  • Also, don't get discouraged by the downvote. I downvoted the question and voted to close it because at the moment, it is not up to site standards (you have shown no work you did on your own). If you edit your question so that you show what you tried and how far you got, I will not only remove the downvote, I will add an upvote. Even if the question is closed, you can still edit it, and we will vote to reopen it. – 5xum May 16 '22 at 13:36
  • Thanks for your advises, I will write about my ideas soon. – Justin May 16 '22 at 13:44
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    are you sure it's $\ln|x_i+x_j|$, not $\ln|x_i-x_j|$? – Jean Marie May 16 '22 at 16:42
  • At $n=2$ this is equivalent to $(x_1+x_2)^2\geqslant4|x_1x_2|$ which doesn't hold (consider $x_1\approx-x_2$). – metamorphy May 18 '22 at 05:51
  • I have written a (modified) follow-up question here – Jean Marie May 18 '22 at 17:46

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