Proving a well-known inequality using S.O.S Using $AM-GM$ inequality, it is easy to show for $a,b,c>0$, $$\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \ge 3.$$
However, I can't seem to find an S.O.S form for $a,b,c$ 
$$f(a,b,c) = \frac{a}{b} + \frac{b}{c} + \frac{c}{a} - 3 = \sum_{cyc}S_A(b-c)^2 \ge  0.$$
Update:
Please note that I'm looking for an S.O.S form for $a, b, c$, or a proof that there is no S.O.S form for $a, b, c$. Substituting other variables may help to solve the problem using the S.O.S method, but those are S.O.S forms for some other variables, not $a, b, c$.

Here is another SOS (Shortest)
$$ab^2+bc^2+ca^2-3abc={\frac {a \left( c-a \right) ^{2} \left({b}^{2}+ ac+cb \right) +b
 \left( 2\,ab+{c}^{2}-3\,ac \right) ^{2}}{4\,ab+ \left( c-a \right) ^
{2}}} \geqslant 0$$
