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 $x,y,z \in \mathbb{R}^{+}$
 A: Your third inequality is on page 185 of The Cauchy-Schwarz Master Class by J. Michael Steele. I will paraphrase his argument.

The weighted AM-GM inequality gives
  $$a^2b^3=(ab^4)^{2/3}(a^4b)^{1/3}\leq {2\over 3}ab^4+{1\over 3}a^4b.$$
If we replace $(a,b)$ in turn by the ordered pairs $(x,y)$ and $(y,x)$, then the sum
  of the resulting bounds gives us $x^2y^3+y^2x^3\leq xy^4+x^4y$.

I think that your other inequalities can be proven in a similar way.
I'm not sure how to use  Karamata here.
A: 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...
A: Define $f(x)=e^x$ and for an inequality of degree $n$ change variables so that $(a,b,c) = \log (x^n,y^n,z^n)$.  Then each assertion is that the left hand side, a sum of values of $f$ at some points, is larger than the sum of $f$ values at another set of points obtained from the first one by a double-stochastic transformation (one whose matrix is non-negative with row and column sums equal to 1).  Majorization ordering and double-stochastic ordering are the same so this would be true for any convex function in place of $e^x$.  The exponential function case gives the Muirhead inequality of which the three inequalities in the question are instances.
The first inequality is thus
$$f(a) + f(b) + f(c) \geq f(a/2 + b/4 + c/4) + f(a/4 + b/2 + c/4) + f(a/4 + b/4 + c/2)$$
and the others can be handled the same way.
Karamata inequality is a synonym for majorization.
