Do more generalizations of Schur's inequality exist? 
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?
 A: Now I have solve it, this inequality it is only true for $n=3,n=5$ holds.
For $n=3$ then inequality can write as
$$(a_{1}-a_{2})(a_{1}-a_{3})+(a_{2}-a_{1})(a_{2}-a_{3})+(a_{3}-a_{1})(a_{3}-a_{2})=a^2_{1}+a^2_{2}+a^2_{3}-a_{1}a_{2}-a_{1}a_{3}-a_{2}a_{3}\ge 0$$
it is clear true
for $n=5$,WLOG, we assume that
$$a_{1}\ge a_{2}\ge a_{3}\ge a_{4}\ge a_{5}$$
Lemma 1:
$$I=(a_{1}-a_{2})(a_{1}-a_{3})(a_{1}-a_{4})(a_{1}-a_{5})+(a_{2}-a_{1})(a_{2}-a_{3})(a_{2}-a_{4})(a_{2}-a_{5})\ge 0$$
Proof: because
$$I=(a_{1}-a_{2})[(a_{1}-a_{3})(a_{1}-a_{4})(a_{1}-a_{5})-(a_{2}-a_{3})(a_{2}-a_{4})(a_{2}-a_{5})]$$
since
$$a_{1}-a_{3}\ge a_{2}-a_{3}\ge 0$$
$$a_{1}-a_{4}\ge a_{2}-a_{4}\ge 0$$
$$a_{1}-a_{5}\ge a_{2}-a_{5}\ge 0$$
$$a_{1}-a_{2}\ge 0$$
so
$$I\ge 0$$
Use same methods we have
$$(a_{5}-a_{1})(a_{5}-a_{2})(a_{5}-a_{3})(a_{5}-a_{4})+(a_{4}-a_{1})(a_{4}-a_{2})(a_{4}-a_{3})(a_{4}-a_{5})\ge 0$$
and
$$(a_{3}-a_{1})(a_{3}-a_{2})(a_{3}-a_{4})(a_{3}-a_{5})\ge 0$$
so for $n=5$ is hold
for $n=4$, you can take $a_{1}=0,a_{2}=a_{3}=a_{4}=1$
for $n\ge 6$,you can take
$$a_{1}=0,a_{2}=a_{3}=a_{4}=1,a_{5}=a_{6}=\cdots=a_{n}=-1$$
then $$\sum_{i=1}^{n}\left(\prod_{j\neq i}(a_{i}-a_{j})\right)=(-1)^3<0$$
A: Here are some inequalities:


*

*Schur's inequality: If $a,b,c,p \geq 0$, then $$a^p(a-b)(a-c)+b^p(b-c)(b-a)+c^p(c-b)(c-a) \geq 0$$

*Weak 6-variable Schur: Suppose $a \geq b \geq c$, $x+z \geq y \geq 0$. Then $$x(a-b)(a-c)+y(b-c)(b-a)+z(c-b)(c-a) \geq 0$$

*Strong 6-variable Schur: Suppose $a \geq b \geq c$, $x+z \geq y \geq 0$. Then $$x^2(a-b)(a-c)+y^2(b-c)(b-a)+z^2(c-b)(c-a) \geq 0$$


Now we get to some inequalities with functions inside it. $f: \mathbb R \rightarrow \mathbb R^+$ be a function that can be expressed as the sum of monotonic non-negative functions. Let $g: \mathbb R \rightarrow \mathbb R$ and $h: \mathbb R \rightarrow \mathbb R$ be odd (functions such that $f(x)=-f(-x)$) and increasing functions. 


*

*Vornicu-Schur inequality: Let $a,b,c,x,y,z \in \mathbb R$, $a \geq b \geq c$ and either $x \geq y \geq z$ or $z \geq y \geq x$. Let $k$ be a postive integer. Then  $$f(x)(a-b)^k(a-c)^k+f(y)(b-a)^k(b-c)^k+f(z)(c-a)^k(c-b)^k \geq 0 $$


We get Schur's inequality if $k=1$, $x=a$, $y=b$, $x=c$, $f(n)=n^p$. But we can generalize further.


*

*Weak generalized Schur inequality: Let $f,g,h$ be defined as above and let $a,b,c \in \mathbb R$, $a \geq b \geq c$. Then $$f(a) g(h(a-b)h(a-c))+ f(b) g(h(b-a)h(b-c))+ f(c) g(h(c-a)h(c-b)) \geq 0$$

*Strong generalized Schur inequality: Let $f,g,h$ be defined as above and let $a,b,c \in \mathbb R$, $a \geq b \geq c$. We further require $f$ to be convex. Then $$f(a)^2 g(h(a-b)h(a-c))+ f(b)^2 g(h(b-a)h(b-c))+ f(c)^2 g(h(c-a)h(c-b)) \geq 0$$

