SolJuly2206

$\mbox{Show that for all nonzero reals }a,b,c,\,$

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

Proof:

$\mbox{Without loss of generality take: }\frac{a}{b}\ge\frac{b}{c}\ge\frac{c}{a} \mbox{ and apply the }$

rearrangement inequality (http://mcraefamily.com/MathHelp/BasicNumberIneqRearrangementInequality.htm).

$\frac{a^2}{b^2}+ \frac{b^2}{c^2} + \frac{c^2}{a^2}\ge\frac{a}{b}\frac{b}{c}+\frac{b}{a}\frac{a}{c}+\frac{c}{b}\frac{b}{c}=\frac{a}{c} + \frac{c}{b} + \frac{b}{a}$

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