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

Note by Joshua Ong 3 years, 3 months ago

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

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

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).

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TopNewestUse AM-GM.

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

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

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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).

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