Prove that if :

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- Feb 20th 2008, 09:21 AMjames_bondInequality
Prove that if :

- Feb 21st 2008, 07:17 AMwingless
The question says ..

- Feb 22nd 2008, 12:35 PMJaneBennet
Suppose one of is 0. Then the other two can’t be 0 if the condition is to hold. Say, we have . Then and the inequality becomes

The LHS, as a quadratic expression in , has negative discriminant and so the inequality is true.

So now we’ll assume . Let . Then and the inequality becomes

Since are positive and , we can make the following substitutions:

where . Then observe that

Hence

So it comes down to proving this inequality:

- Feb 22nd 2008, 09:21 PMCharbel
hmmm... don’t know why I was heavily neg repped on this question =\ but yer .. anyways...

^^ top was interesting

After about 2 pages of working trying to find similarities between the two I got this...

LHS

8(a+b+c)^2

8a^2 + 16ab + 16ac + 8b^2 + 16bc + 8c^2

8(a^2 + b^2 + c^2) + 16(ac + ab + cd)

8(a^2 + b^2 + c^2) + 16(3)

8(a^2 + b^2 + c^2) + 48

8(a^2 + b^2 + c^2 + 6)

RHS

9(a+b)(a+c)(b+c)

9a^2b + 9a^2c + 9ab^2 + 18abc + 9ac^2 + 9b^2c + 9bc^2

9a(ab + ac) + 9b(ab + bc) + 9c(ac + bc) +18(abc)

9a(3-bc) + 9b(3-ac) + 9c(3-ab) +18abc

27a – 9abc + 27b – 9abc + 27c -9abc + 18abc

27(a + b + c) – 9abc

9(3(a + b + c) - abc)

So we can say….

8(a^2 + b^2 + c^2 + 6) => 9(3(a + b + c) - abc)

when ab + cb + bc = 3…..

I also noticed a + b + c appears In the RHS which also appears in the LHS

So we can let a + b + c = d

And say:

8d^2 => 27d – 9abc

still doesn’t help unless we know a b or c = 0…