High powered Inequality

Prove $x^4+y^4+z^4\geq xyz(\sqrt{xy}+\sqrt{yz}+\sqrt{xz})$.

Note by Joshua Ong
3 years, 11 months ago

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Use AM-GM.

- 3 years, 11 months ago

The result follows immediately from Muirhead: $$(4,0,0)\succ \left(\dfrac{3}{2},\dfrac{3}{2},1\right)$$

- 3 years, 11 months ago

Except it is not given that $$x,y,z>0$$ and Muirhead's inequality holds for positive reals.

- 3 years, 11 months ago

Are $$x,y,z$$ positive reals or just reals? (the same question for some of the other problems. You should provide the given restrictions on $$x,y,z$$ since otherwise we're not quite sure what they are).

- 3 years, 11 months ago