<s>[INST] Inequality - Cauchy Schwarz Let $a, b, c, d > 0 \in \mathbb{R}$ such that $a^2 + b^2 + c^2 + d^2 = 4$. Show that:
$S = \frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{d} + \frac{d^2}{a} \geq 4$
My approach:
I used the Cauchy-Schwarz inequality to show that $S \geq a + b + c + d$ but that is useless as $a + b + c + d \leq 4$. How would you approach this problem? (Only hints desired)
 [/INST] By Holder's Inequality,
$$S^2\sum_{cyc} a^2b^2\ge (a^2+b^2+c^2+d^2)^3$$
So it remains to note that by AM-GM
$$\sum_{cyc}a^2b^2 = (a^2+c^2)(b^2+d^2) \le \frac{\left((a^2+c^2)+(b^2+d^2)\right)^2}4$$
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