Inequalities of high level

Question

For positive reals $a,b,c$ prove $$\frac{a+b-2c}{b+c} +\frac{b+c-2a}{c+a} + \frac{c+a-2b}{a+b}\geq 0$$

I proved this if these $a,b,c$ were sides of a triangle but could not proceed further

What is Ravi's substitution?pls clarify it and then I hope I can solve this question — Caratheodory_Enthusiast, Mar 07 '15 at 13:51
can u write some important points of this inequality. I have 2g internet connection which is opening this link very slow — Caratheodory_Enthusiast, Mar 07 '15 at 14:15
@G-man The substitution you mention would have been useful to prove if $a, b, c$ where sides of a triangle. For all positive reals, it is not at all obvious how that would work. — Macavity, Mar 08 '15 at 16:24

score 3 · Answer 1 · answered Mar 08 '15 at 07:00

3

Hint: Note that $$\prod_{cyc}(b+c) \sum_{cyc}\frac{a+b-2c}{b+c} = \sum_{cyc} a(a-c)^2$$

answered Mar 08 '15 at 07:00

Macavity

1 Answers1