Let $F(a,b,c,r)=(P(a,b,c)/Q(a,b,c))^r$ where $P(a,b,c)$ and $Q(a,b,c)$ are homogenous polynomials with equal degrees $deg(p)=deg(q)=n$, defined for non-negative $a,b$, and $c$.
Below, we assume everywhere (for simplicity) that $a+b+c=3$.
In his "Secrets of inequalities" (vol.2, p.323) Pham Kim Hung proves a result equivalent to the folloiwng:
If $P$ and $Q$ are linear ($n=1$) and $r=1$ then the minimum of $F(a,b,c)+F(b,c,a)+F(c,a,b)$ is achieved either at (1,1,1) or at the boundary of the region (e.g. $c=0$).
I would like to determine under what conditions this can be generalized for other values of $n$ and $r$. If you look at the Inequality tag here on StackExchange, you'll find lots of examples of inequalities which reduce to this cyclic sum for some well-chosen $F$ and for which the extremum is at $(1,1,1)$ or the boundary. This is of course very much inderstandable given Lagrange multiplier theorem.
Here is an example ($a+b+c=3$):$$ \left(\frac{a^2 + b c}{b^2 + a c}\right)^r + \left(\frac{b^2 + a c}{a b + c^2}\right)^r + \left(\frac{a b + c^2}{a^2 + b c}\right)^r \geq 3$$
The minimum in this case is achieved both at $(1,1,1)$ and at $(1.5,1.5,0)$
I would like to propose two challenges.
Challenge 1 Find an example of $P$ and $Q$ with lowest possible degree, for which the minimum of $F(a,b,c)+F(b,c,a)+F(c,a,b)$ for $r=1$ is achieved in a non-trivial point - meaning not $(1,1,1)$ and not $abc=0$.
Challenge 2 Find an example of $P$ and $Q$ with lowest possible degree, for which the minimum of $F(a,b,c,1)+F(b,c,a,1)+F(c,a,b,1)$ is achieved in a trivial point, but there exists $r>0$ for which the minimum of $F(a,b,c,r)+F(b,c,a,r)+F(c,a,b,r)$ is achieved in a non-trivial point.