<s>[INST] What is the relationship between $\frac{1}{3}x_1+\frac{1}{3}x_2+\frac{1}{3}x_3$ and $x_1^{\frac{1}{3}}x_2^{\frac{1}{3}}x_3^{\frac{1}{3}}$? Is $\frac{1}{3}x_1+\frac{1}{3}x_2+\frac{1}{3}x_3$ > or < or = $x_1^{\frac{1}{3}}x_2^{\frac{1}{3}}x_3^{\frac{1}{3}}$, given $x_1$, $x_2$ and $x_3$ are all positive?
I know there exists a real number $a\in(1,2)$ such that $\frac{1}{3}x=x^{\frac {1}{3}}$, but there are three variables even though they are all positive.
Also I know how to simplify $\frac{1}{3}x_1+\frac{1}{3}x_2+\frac{1}{3}x_3-x_1^{\frac{1}{3}}x_2^{\frac{1}{3}}x_3^{\frac{1}{3}}$, but then I get stuck too.
I also tried to relate this to the roots of polynomial, but I don't quite know how.
Could anyone help please?
 [/INST] For what it's worth, you can prove this inequality without knowing the more general result: write $x_i=y_i^3$ so$$\frac{y_1^3+y_2^3+y_3^3}{3}-y_1y_2y_3=\tfrac16(y_1+y_2+y_3)((y_1-y_2)^2+(y_2-y_3)^2+(y_3-y_1)^2).$$
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