<s>[INST] If $a,b,c$ are distinct positive numbers , show that $\frac{a^8 + b^8 + c^8}{a^3 b^3 c^3} > \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ If $a,b,c$ are distinct positive numbers , show that $$\frac{a^8 + b^8 + c^8}{a^3 b^3 c^3} > \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$$ I am thinking of Tchebycheff's inequality for this question, but not able to proceed. How do I solve this?
 [/INST] This is the same as
$$a^8+b^8+c^8> a^2b^3c^3+a^3b^2c^3+a^3b^3c^2.$$
You can get this by using AM/GM to get an upper bound for each term
on the right.
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