Proof Using Rearrangement Inequality

[James Brady]

The Rearrangement Inequality states that for two sequences ${a_i}$ and ${b_i}$, the sum $S_n = \sum_{i=1}^n a_ib_i$ is maximized if $a_i$ and $b_i$ are similarly arranged. That is, big numbers are paired with big numbers and small numbers are paired with small numbers.

The question given is using the above theorem to to prove that for any given three positive whole numbers a,b and c:

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

Thinking so far: the theorem deals with maximizing or minimizing the Cartesian product of sets, but we have denominators in each term, maybe I can rewrite the denominators using negative exponents...? But I don't see how that gets me any closer either, basically I need some kind of nudge in the right direction.​

 

[micromass]

Hint:
$$\frac{b}{a}+\frac{a}{c} + \frac{c}{b} = \frac{b}{c}\frac{c}{a}+\frac{a}{b}\frac{b}{c}+\frac{a}{b}\frac{c}{a}$$

 

[James Brady]

Perfect, I used that to build a common denominator which then canceled out.