How to prove this high-degree inequality Let $x$,$y$,$z$ be positive real numbers which satisfy $xyz=1$. Prove that:
$(x^{10}+y^{10}+z^{10})^2 \geq 3(x^{13}+y^{13}+z^{13})$.
And there is a similar question: Let $x$,$y$,$z$ be positive real numbers which satisfy the inequality
$(2x^4+3y^4)(2y^4+3z^4)(2z^4+3x^4) \leq(3x+2y)(3y+2z)(3z+2x)$. Prove this inequality:
$xyz\leq 1$.

Another way. By my previous post it's enough to prove that:
$$(x^5+y^5+z^5)^2\geq3xyz(x^7+y^7+z^7)$$ for positive $x$, $y$ and $z$.
Indeed, let $x^5+y^5+z^5=\text{constant}$ and $x^7+y^7+z^7=\text{constant}$.
Thus, by the Vasc's EV Method (see here: Cîrtoaje - The equal variable method Corollary 1.8(b)) it's enough to prove the last inequality (because it's homogeneous) for $y=z=1$, which gives
$$(x^5+2)^2\geq3x(x^7+2)$$ or
$$(x-1)^2(x^8+2x^7-2x^5-4x^4-2x^3+2x+4)\geq0$$ and the rest is smooth.
