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...