simple inequality

by hemangsarkar, Sep 1, 2012, 6:28 PM

Q) if $x,y,z$ are real positive numbers, such that $x + y + z = 2$ , then prove that -

$\sum_{cyclic}x^4 \geq \frac{16}{27}$

my solution :

by cauchy schwarz inequality we have,
$\sum_{cyclic}x^4 \geq \frac{(\sum x^2)^2}{3} \geq \frac{(\sum x)^4}{27} = \frac{16}{27}$

equality holds if $x = y = z = \frac{2}{3}$ .

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How about applying Holder's inequality Which would Simplify This
$\left(x^4+y^4+z^4\right)\left(1+1+1\right)\left(1+1+1\right)\left(1+1+1\right)\ge\left(x+y+z\right)^4$
But Any way would be easy for this one..

by boywholived, Sep 8, 2012, 12:01 PM

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simple inequality

by hemangsarkar, Sep 1, 2012, 6:28 PM

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