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An Easy Inequality

For all non-negative reals \(x,y,z\), prove that \[x^2+y^2+z^2\ge x\sqrt{yz}+y\sqrt{zx}+z\sqrt{xy}\]

Note by Daniel Liu
3 years, 1 month ago

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While Muirhead works, another viable option is a generalization of rearrangement by taking four copies of \(\{\sqrt{x},\,\sqrt{y},\,\sqrt{z}\}\). (WLOG \(x\geq y\geq z\geq 0\)) Since \(\{\sqrt{x},\,\sqrt{y},\,\sqrt{z}\}\) is monotonic decreasing, we have

\[\sqrt{x}\cdot\sqrt{x}\cdot\sqrt{x}\cdot\sqrt{x} + \sqrt{y}\cdot\sqrt{y}\cdot\sqrt{y}\cdot\sqrt{y} + \sqrt{z}\cdot\sqrt{z}\cdot\sqrt{z}\cdot\sqrt{z}\] \[\geq \sqrt{x}\cdot\sqrt{x}\cdot\sqrt{y}\cdot\sqrt{z} + \sqrt{y}\cdot\sqrt{y}\cdot\sqrt{z}\cdot\sqrt{x} + \sqrt{z}\cdot\sqrt{z}\cdot\sqrt{x}\cdot\sqrt{y}\] Bob Krueger · 3 years, 1 month ago

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A solution by Muirhead that I discovered while writing the problem is as follows:

Note that \((2,0,0)\succ \left(1,\dfrac{1}{2},\dfrac{1}{2}\right)\).

Thus, by Muirhead, \[2(x^2+y^2+z^2)\ge 2(x\sqrt{yz}+y\sqrt{zx}+z\sqrt{xy})\] and the result follows.

The challenge I give you is to try to prove it using AM-GM. It shouldn't be too difficult. Daniel Liu · 3 years, 1 month ago

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@Daniel Liu By AM-GM,

\[\dfrac{x^2 + x^2 + y^2 + z^2}{4} \ge \sqrt[4]{x^4y^2z^2}\]

so \[\dfrac12x^2 + \dfrac14y^2 + \dfrac14z^2 \ge x\sqrt{yz}.\] Similarly, if we duplicate the \(y^2\) and \(z^2\) terms, we get

\[\dfrac14x^2 + \dfrac12y^2 + \dfrac14z^2 \ge y\sqrt{zx},\] and

\[\dfrac14x^2 + \dfrac14y^2 + \dfrac12z^2 \ge z\sqrt{xy}.\] Adding these inequalities together, we have

\[x^2+y^2+z^2 \ge x\sqrt{yz} + y\sqrt{zx} + z\sqrt{xy}\] as requested. \(\square\) Michael Tang · 3 years, 1 month ago

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@Michael Tang Ah, you beat me to it with the same solution I posted. Kudos to you on being first. Trevor B. · 3 years, 1 month ago

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@Daniel Liu By AM-GM, \(\dfrac{2x^2+y^2+z^2}{4}=\dfrac{x^2+x^2+y^2+z^2}{4}\ge\sqrt[4]{x^4y^2z^2}.\)

Similarly, \(\dfrac{x^2+2y^2+z^2}{4}\ge\sqrt[4]{x^2y^4z^2}\) and \(\dfrac{x^2+y^2+2z^2}{4}\ge\sqrt[4]{x^2y^2z^4}\). Note that \(\sqrt[4]{x^4y^2z^2}=\sqrt{x^2yz}=x\sqrt{yz},\) which can also apply to the other three radicals.

Dividing through, we have this system of inequalities.

\[ \begin{align} \dfrac{1}{2}x^2+\dfrac{1}{4}y^2+\dfrac{1}{4}z^2&\ge x\sqrt{yz}\\ \dfrac{1}{4}x^2+\dfrac{1}{2}y^2+\dfrac{1}{4}z^2&\ge y\sqrt{zx}\\ \dfrac{1}{4}x^2+\dfrac{1}{4}y^2+\dfrac{1}{2}z^2&\ge z\sqrt{xy} \end{align} \]

Adding these together yields \(x^2+y^2+z^2\ge x\sqrt{yz}+y\sqrt{zx}+z\sqrt{xy},\) as desired.

\[\mathbb{Q.E.D.}\]


@Daniel Liu, Can you please explain your Muirhead solution a little further? I'm not familiar with that. Trevor B. · 3 years, 1 month ago

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@Trevor B. Muirhead states that for any two sets of numbers \(a_1,a_2,\ldots a_n\) and \(b_1,b_2,\ldots b_n\), if \((a_1,a_2,\ldots a_n)\succ(b_1,b_2,\ldots b_n)\), then \[\sum_{sym}x_1^{a_1}x_2^{a_2}\cdots x_n^{a_n}\ge \sum_{sym}x_1^{b_1}x_2^{b_2}\cdots x_n^{b_n}\]

Search up Majorization if you do not know what \(\succ\) (which means majorizes) means.

But Muirhead is basically just a generalized AM-GM. Setting the first set of numbers as \(1,0,\ldots 0\) and the second set as \(\dfrac{1}{n},\dfrac{1}{n},\ldots \dfrac{1}{n}\), we see that \((1,0,\ldots 0)\succ \left(\dfrac{1}{n},\dfrac{1}{n},\ldots \dfrac{1}{n}\right)\) and using Muirhead's gives us the familiar \(n\)-term AM-GM. Daniel Liu · 3 years, 1 month ago

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We could apply AM-GM on \(\sqrt {yz}\le \frac {y+z}{2}\) which would lead to proving the well known \(x^2+y^2+z^2\ge xy+yz+xz\) Xuming Liang · 3 years, 1 month ago

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@Xuming Liang It is also true that \[xy+yz+zx\ge x\sqrt{yz}+y\sqrt{zx}+z\sqrt{xy}\]

Can you prove that? Daniel Liu · 3 years, 1 month ago

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@Daniel Liu Well by doing what I said this is actually what you get :).....: \(x\sqrt {yz}+y\sqrt {zx}+z\sqrt {xy}\le x\frac {y+z}{2}+ y\frac {x+z}{2}+z\frac {y+x}{2}=xy+yz+zx\) Xuming Liang · 3 years, 1 month ago

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@Xuming Liang Ah, I misunderstood your statement. Sorry.

However, it is also true that \[(x\sqrt{yz}+y\sqrt{zx}+z\sqrt{xy})^3\ge 27x^2y^2z^2\]

How about that one? Daniel Liu · 3 years, 1 month ago

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@Daniel Liu No need to apologize :) Anyway this one is just immediate AM-GM Xuming Liang · 3 years, 1 month ago

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@Xuming Liang It's good to finally see some inequality actions going on here, I've never been interested in inequalities until recently :) Here's one I feel kinda proud for solving it:

\(a,b,c\) are positive reals, \(ab+bc+ca=1\) prove \( \sqrt[3]{\frac{1}{a}+6b}+\sqrt[3]{\frac{1}{b}+6c}+\sqrt[3]{\frac{1}{c}+6a }\leq\frac{1}{abc}. \)

P.S. Good think on AoPS there's a quick way for you to copy and paste latex.. Xuming Liang · 3 years, 1 month ago

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@Xuming Liang lol yea no need to do any clever separating of terms.

How about this one?

\[2x\sqrt{yz}+2y\sqrt{zx}+2z\sqrt{xy}\ge \sum_{sym}\sqrt[3]{xy^2z^3}\] Daniel Liu · 3 years, 1 month ago

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@Daniel Liu @Michael Tang @Trevor B. Try your hand at this one. Daniel Liu · 3 years, 1 month ago

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@Daniel Liu I'm neither Michael ,nor Trevor, but since

\(1,\frac{1}{2},\frac{1}{2}\) majorizes \(1,\frac{2}{3},\frac{1}{3}\), use Muirhead to obtain the result. Bogdan Simeonov · 3 years, 1 month ago

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@Bogdan Simeonov Sorry, but the point was to prove it using AM-GM @Bogdan Simeonov

Since anything provable using Muirhead (which was the instakill solution that I suggested) could be proven using AM-GM, the challenge was to prove my suggested inequalities using AM-GM. Daniel Liu · 3 years, 1 month ago

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@Daniel Liu Oh, ok :D Bogdan Simeonov · 3 years, 1 month ago

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