<s>[INST] Half-symmetric, homogeneous inequality Let $x,y,z$ be three positive numbers.  Can anybode prove the follwing inequality :
$(x^2y^2+z^4)^3 \leq (x^3+y^3+z^3)^4$ (or find a counterexample, or find a reference ...)
 [/INST] Note that we have $(x^3+y^3+z^3)^2\geq (x^3+y^3)^2+z^6\geq 4x^3y^3+z^6,$ and also $x^3+y^3\geq 2\sqrt{x^3y^3},$ so that it suffices to check that
$$\left(2\sqrt{x^3y^3}+z^3\right)^2(4x^3y^3+z^6)\geq (x^2y^2+z^4)^3.$$
Using the Holder's inequality, we can get a sharper bound:
$$\begin{aligned}\left(2\sqrt{x^3y^3}+z^3\right)^2(4x^3y^3+z^6)&\geq \left(\sqrt[3]{16x^6y^6}+z^4\right)^3\\&=\left(2\sqrt[3]2x^2y^2+z^4\right)^3.\end{aligned}$$
In this stronger version, equality holds for $(x,y,z)=(t,t,\sqrt[3]{2t}).$
For the original inequality, if nonnegative reals are allowed, then equality holds if and only if $x=y=0.$ Otherwise there is no equality.
$\Box$
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