I've noticed that this site gets a lot of very similar questions about the cyclic sum of three terms. For example,
- Prove $\frac{a\sqrt{b+bc+c}}{b+c}+\frac{b\sqrt{c+ca+a}}{c+a}+\frac{c\sqrt{a+ab+b}}{a+b}\ge 2$ for $a+b+c=2$.
- $a,b,c>0:a+b+c=3$. Prove that: $\sum\sqrt{\frac{ab+2}{ab+c}}\ge \frac{3\sqrt{6}}{2}$.
Often, the holder-inequality will be mentioned in the question or answers.
I'm wondering if there's a general solution to the problem. That is, given a function $f$ of three real variables, and a constraint that those variables are positive and have a specified fixed sum $K$, find the minimum or maximum value of $g(a, b, c) := f(a, b, c) + f(b, c, a) + f(c, a, b)$, subject to $a+b+c=K$.
You may assume that $f$ is continuous, differentiable, or otherwise “reasonable”.
