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?

  • 2
    $\begingroup$ I think that your work by rearrangement is correct. $\endgroup$ – chloe_shi Jun 25 '14 at 5:43

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

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.