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I know that the AM-GM inequality takes the form $$ \frac{x + y}{2} \geq \sqrt{xy},$$ but I read in a book another form which is $$ \frac{x^2 + y^2}{2} \geq |xy|,$$ but I am wondering how the second comes from the first? could anyone explain this for me, please?

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If you plug $x=X^2$, $y=Y^2$ into the first inequality you get $$\frac{X^2+Y^2}{2} \ge \sqrt{X^2Y^2} = \sqrt{(XY)^2}=|XY|,$$ which is the second inequality (modulo capitalization).

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The AM-GM inequality for $n$ non-negative values is

$\frac1{n}(\sum_{k=1}^n x_k) \ge (\prod_{k=1}^n x_k)^{1/n} $.

This can be rewritten in two ways.

First, by simple algebra,

$(\sum_{k=1}^n x_i)^n \ge n^n(\prod_{k=1}^n x_k) $.

Second, letting $x_k = y_k^n$, this becomes

$\frac1{n}(\sum_{k=1}^n y_k^n) \ge \prod_{k=1}^n y_k $.

It is useful to recognize these disguises.

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