Prove that `(a^8+b^8+c^8)/(a^3b^3c^3)>1/a+1/b+1/c`

Explanation

We have to prove that
`a^(8) + b^(8) + c^(8) gt a^(2) b^(2) c^(2) (bc + ca + ab)`. Now
`(a^(8) + b^(8) + c^(8))/(3) gt ((a + b + c)/(3))^(8)`
or `(a^(8) + b^(8) + c^(8))/(3) gt ((a + b + c)/(3))^(6) ((a + b + c)/(3))^(2) gt [(abc)^(a//3)]^(6) [(a^(2) + b^(2) + c^(2) + 2ab + 2bc + 2ca)/(9)]`
`( :' A.M gt G.M)`
But `a^(2) + b^(8) + c^(2) gt ab + bc + ca`
So, `(a^(8) + b^(8) c^(8))/(3) gt a^(2) b^(2) c^(2) ((3ab + 3bc + 3ca))/(9))`
or `a^(8) + b^(8) + c^(8) gt a^(2) b^(2) c^(2) (ab + bc + ca)`
or `(a^(8) + b^(8) + c^(8))/(a^(3)b^(3)c^(3)) gt (ab + bc + ca)/(abc)`
or `(a^(5) + b^(8) + c^(8))/(a^(3) b^(3) c^(3)) gt (1)/(a) + (1)/(b) + (1)/(c )`