Let real numbers $a,b,c>0$ such that $a+b+c=1$ and $ab+bc+ca=q$, where $q>0$. We consider the following expression: $$S_1=\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}.$$ Find the minimum (or infimum, if exists) of $S_1$ in 2 cases: $q=\dfrac{1}{4}$ and $q=\dfrac{8}{27}$.
Open problem: Find a boundary of $q$ so that $S_1$ has minimum (or infimum).
For a long time, there have been many inequality problems involving cyclic expressions like $$S_0=\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}.$$ While we can handle the boundary of $S_0$ (see https://artofproblemsolving.com/community/c6h183680), $S_1$ seems being more difficult than $S_0$.
Hence I make this discussing post and see if it's actually possible to do the same.