Prove the inequality: $(y+z)*x^4+(x+z)*y^4+(x+y)*z^4\le((x+y+z)^5)/12$ for positive reals. The inequality is homogeneous, but I am not sure if that helps.
Also, I was thinking about considering a concave function and apply Jensen or Karamata on LHS.

Suppose $x \geqslant y \geqslant z$ and
$$f(x,y,z) = x^4(y+z)+y^4(z+x)+z^4(x+y).$$
We have
$$\begin{aligned}f(x,y,z)-f(x,y+z,0) & = y^4(z+x)+z^4(x+y)-x(y+z)^4\\ \\& = y^4(z+x)+z^4(x+y)-x(y+z)^4 \\ \\&=-yz\left[3x(y+z)^2+y^2(x-y)+z^2(x-z)\right] \leqslant 0.\end{aligned}.$$
Therefore $f(x,y,z) \leqslant f(x,y+z,0).$ Finally, we need to prove
$$f(x,y+z,0) = x^4(y+z)+x(y+z)^4\leqslant \frac{1}{12}(x+y+z)^5.$$
Setting $a = y +z,$ the inequality become
$$12ax(a^3+x^3) \leqslant (x+a)^5,$$
or
$$(a+x)(a^2-4ax+x^2)^2 \geqslant 0.$$
Which is true. The proof is completed.
