It looks so simple, but...

[Arejang]

This is off a mathematics journal; it's an inequality that seems so simple I want to see if any of you guys have any better luck at it than I did.

Show that for any positive real numbers a, b, and c.


abc is less than or equal to {(a+b+c)^3 - (a^3+b^3+c^3)}/24 is less than or equal to (1/27)*(a+b+c)^3

Let me know if the equation is unclear. I will appreciate any help given, thanks!

[HallsofIvy]

Looks like the "harmonic-geometric-arithmetic mean" inequality for n= 3.

For n numbers, the arithmetic mean is the sum of the numbers divided by n.
The geometric mean is the nth root of the product of the numbers.
The harmonic mean is the reciprocal of the arithmetic mean of the the reciprocals of the numbers.

It can be shown that for any set of n numbers,
harmonic mean<= geometric mean<= arithmetic mean which, with a little algebra, reduces to your inequality.