Help on simple-looking ineqality

1. Apr 6, 2006

phonic

Dear all,

I come across to a simple-looking ineqality. But I cann't prove it for quite a long time. Could any body give a hint? Thanks a lot!

$$2[(n-1) \sum_{j=1}^n r_j^2 -(n-2) r_n^2] \geq (\sum_{j=1}^n r_j)^2$$

where $$n\geq 2, \forall r_j \geq 0, j=1,2,\cdots,n$$.

2. Apr 6, 2006

HallsofIvy

Staff Emeritus
Are you saying that you do not understand why (n-3)2rn2 should be non-negative?

Last edited: Apr 7, 2006
3. Apr 7, 2006

phonic

Thanks for your sugestions. However, I still don't know how did you get to the simpification of $$(n-3)^2r_n^2$$.

According to your sugestions, I got the following proof:

$$2[(n-1)\sum_{j=1}^n r_j^2 -(n-2) r_n^2] =2[(n-1)\sum_{j=1}^{n-1} r_j^2 + r_n^2] \geq 2[(\sum_{j=1}^{n-1} r_j)^2 + r_n^2] \geq (\sum_{j=1}^{n-1} r_j)^2 + r_n^2 + 2r_n \sum_{j=1}^{n-1} r_j =(\sum_{j=1}^{n} r_j)^2$$