Q) if , then prove that



my solution :
by arithmetic mean - geometric mean inequality, we have -


so we have .. (1)

also,
( by the same am - gm argument).

or,

...(2)

adding (1) and (2), we get

$\frac{a^6 + b^6 + c^6 + 15}{12} -\frac{3}{a^6 + b^6 + c^6 + 3} \geq \frac{(abc)^2 + 5}{4} -\frac{3}{3(abc)^2 + 3}$

here we take .

then we have to prove that


which is equivalent to

or,
or,


which is true.
hence proved.

check the equality case on your own.