Inequality proof: 3 (a/b + b/a + b/c + c/b + a/c + c/a) +...

[LanNguyen]

Help is needed to prove the following 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)

for all a, b, c are positive reals
-------------------------
If it helps, I know the equality occurs when a=b=c=2 (although I'm not sure if it's the only one).

If we can prove (2, 2, 2) is the only point in the positive real domain at which equality occurs then we can prove the above inequality by showing the Hessian matrix at this point is positive definite.

However, given the symmetry of the inequality, I believe there would be a simpler proof based on clever algebraic manipulation and fundamental inequalities. Can any one give it a try either ways,pls?

Any help/hints is appreciated...

 

[Deleted member 4993]

Help is needed to prove the following 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)

for all a, b, c are positive reals
-------------------------
If it helps, I know the equality occurs when a=b=c=2 (although I'm not sure if it's the only one).

If we can prove (2, 2, 2) is the only point in the positive real domain at which equality occurs then we can prove the above inequality by showing the Hessian matrix at this point is positive definite.

However, given the symmetry of the inequality, I believe there would be a simpler proof based on clever algebraic manipulation and fundamental inequalities. Can any one give it a try either ways,pls?

Any help/hints is appreciated...

I have not tried it yet - but relevant equality comes to mind:

\(\displaystyle a^3 \, + \, b^3 \, + \, c^3 \, - \, 3abc \, = \, (\, a\, + \, b\, + \, c\,)\cdot(\, a^2\, + \, b^2\, + \, c^2\, - \, ab\, - \, bc\, - \, ca\,)\)