<s>[INST] $\frac{a}{b}+ \frac{b}{c} + \frac{c}{a} \geq \frac{9(a^2+b^2+c^2)}{(a+b+c)^2}$ Prove the following inequality
$$\frac{a}{b}+ \frac{b}{c} + \frac{c}{a} \geq \frac{9(a^2+b^2+c^2)}{(a+b+c)^2}, \enspace \forall a,b,c \in (0,\infty)$$
I tried by multying both sides by the denominator $(a+b+c)^2$ and then applying Holder for the left side but I couldn’t work it out. I  would prefer a proof without never ending computations (a.k.a. Brute Force / opening up the brackets) because I know how to do it this way already. A proof using C-B-S, Holder, Titu’s Lemma or their generalizations or other well-known inequalities would be ideal. Thank you!
 [/INST] There is also the following proof (not mine) by SOS.
$$\frac{a}{b}+ \frac{b}{c} + \frac{c}{a}-\frac{9(a^2+b^2+c^2)}{(a+b+c)^2}=\frac{\sum\limits_{cyc}\left((a^2-ab)abc+a(a-b)^2(b-2c)^2\right)}{abc(a+b+c)^2}\geq0.$$
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