For $x,y,z$ positive numbers such that $xyz = 1$, prove that: $$2 \sqrt{2} (x^7+y^7)(x^7+z^7)(y^7+z^7) \geq \sqrt{(x^{16}+7)(y^{16}+7)(z^{16}+7)}$$

Solution

$$2(x^7+y^7)(x^7+z^7) = x^{14} + (x^{14} + 2x^7(y^7+z^7) + 2y^7z^7)$$ by AM-GM: $$\geq x^{14} + 7x^6y^4z^4 = \frac{x^{16}+7}{x^2}$$ By multiplying similar inequalities and taking square root, we get the desired result