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Prove that : If a,b,c $\in \mathbb{R^+}$

$$\frac a{b^2} + \frac b{c^2} +\frac c{a^2} \geq \frac1a+\frac1b+ \frac1c$$

My attempt :

We know that the sequence {a,b,c} and {$\frac1{a^2},\frac1{b^2},\frac1{c^2}$} are oppositely ordered thus from rearrenegement inequality we conclude -

$$\frac a{b^2} + \frac b{c^2} +\frac c{a^2} \geq \frac{a}{a^2}+\frac{b}{b^2}+ \frac{c}{c^2}$$

Is this correct?

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    $\begingroup$ I think that your work by rearrangement is correct. $\endgroup$
    – chloe_shi
    Commented Jun 25, 2014 at 5:43

1 Answer 1

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$\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\left(\dfrac{a}{b^{2}}+\dfrac{b}{c^{2}}+\dfrac{c}{a^{2}}\right)\geq\left(\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{a}\right)^{2}$ : Cauchy-Schwarz

$\therefore$ $\dfrac{a}{b^{2}}+\dfrac{b}{c^{2}}+\dfrac{c}{a^{2}}\geq \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$

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