<s>[INST] Advice on using the Karamata Inequality Is it possible to use the Karamata inequality to prove the following inequalities to be true ?

$$x^{4}+y^{4}+z^{4}\geq x^{2}yz + xy^{2}z + xyz^{2}$$
  $$x^{5}+y^{5}\geq x^{3}y^{2} + x^{2}y^{3}$$
  $$x^{4}y + xy^{4}\geq x^{3}y^{2} + x^{2}y^{3}$$

for all $x,y,z \in \mathbb{R}^{+}$
 [/INST] As Byron observed all your inequalities follow immediately from AM-GM inequality.
To prove the first one, you can do the following:
$$\frac{x^4+x^4+y^4+z^4}{4} \geq x^2yz \,$$
$$\frac{x^4+y^4+y^4+z^4}{4} \geq xy^2z \,$$
$$\frac{x^4+y^4+z^4+z^4}{4} \geq xyz^2 \,$$
and add them together.
The second one follows from
$$\frac{x^5+x^5+x^5+y^5+y^5}{5} \geq x^3y^2$$
$$\frac{x^5+x^5+y^5+y^5+y^5}{5} \geq x^2y^3$$
while the last one 
$$\frac{x^4y+x^4y+xy^4}{3} \geq x^3y^2$$
$$\frac{x^4y+xy^4+xy^4}{3} \geq x^2y^3$$
Actually, all your inequalities are a particular case of the Muirhead's Inequality. Muirhead is similar in idea to Karamata inequality, but I think it is not a consequence of it.... I actually doubt that you can use Karamata inequality here, since you have two sets of variables: $x,y,z$ and the powers.
As for Muirhead, your first inequality is just Muirhead for $[4,0,0] \succeq [2,1,1]$, your second inequality is Muirhead $[5,0] \succeq [3,2]$ and the last is Murihead $[4,1] \succeq [3,2]$.
P.S. Here is a better link for Muirhead Inequality.
Muirhead Explained
P.P.S. The solution above is just the standard "Prove this particular case of Muirhead by using AM-GM" approach...
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