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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 $x,y,z \in \mathbb{R}^{+}$

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