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I meet this following problem If $$n\ge 3,\sum_{i=1}^{n}\left(\prod_{j\neq i}(a_{i}-a_{j})\right)\ge 0$$ where $a_{i}$ are real numbers.

when $n=3$, it is Schur's inequality so which $n$ such this inequality?

but for more generalization form of Schur'.s Inequality exists?

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    $\begingroup$ For $n=4$ choose $a_{1}=10$, $a_{2}=9$, $a_{3}=7$, and $a_{4}=2$ for a counter example for your version of Schur's inequality. However, if one allows for further constraints one can prove something similar to Schur's inequality, see e.g. sciencedirect.com/science/article/pii/S2212017315001152. (Actually I got my example from that paper.) For $n$ even you can choose numbers with the smallest number far below the others to construct a very large negative term. $\endgroup$
    – Johannes Trost
    Commented Jul 1, 2015 at 16:54
  • $\begingroup$ this inequality hold only for $n=3,5$,see my answer:math.stackexchange.com/questions/1346798/… $\endgroup$
    – math110
    Commented Jul 8, 2015 at 15:55

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