Very nice problem! Could you tell me the source of this problem? Thanks.
Pls help to prove this inequality which I think is quite neat:
3 (a/b + b/a + b/c + c/b + a/c + c/a) + (1 + a) (1 + b) (1 + c) (c/b + c/a) (b/a + b/c) (a/b + a/c)
6 abc + 6 + 9(ab + bc + ac + a + b + c) + 3(ab/c + bc/a + ac/b)
with a, b, c are positive reals
If it helps, equality occurs when a=b=c=2. Also, can any one helps to prove (2, 2, 2) is the only point at which equality occur.
I discovered this inequality while working on a mathematical modelling of a genetic system with negative feedback loops. I'm working in systems biology area.
I know for sure the inequality holds and can prove it computationally. I'm looking for a rigorous, mathematical proof though. Have you tried it out? Do you have any ideas on tackling it?
I am not certain if this will be helpful, but I'm starting out with the observation that for a pair of positive reals A and B:
This inequality can be proven simply by writing it as
Since (A - B)^2 will always be at least zero, the inequality holds true.
Hope that helps.
I am now attempting to prove that
It seems to me it is true, but I haven't found a mathematical proof for it yet. This is as far as I've gone:
So it's now a matter of proving that ab(ab - c) + bc(bc - a) + ac(ac - b) is at least zero. Any takers?