The uvw method is a very useful method for the proof of polynomial inequalities with three variables. Sometimes it works for more variables as well. This tag should be used for questions that could be tackled with this method, or questions about the method itself.

Let say we need to prove that $P\geq0$, where $P$ is a symmetrical polynomial over $\mathbb R$ of variables $a$, $b$ and $c$.

Let $a+b+c=3u$, $ab+ac+bc=3v^2$, where $v^2$ can be negative, and $abc=w^3$.

Hence, there is polynomial $f$ with three variables for which $f(u,v^2,w^3)=P(a,b,c)$.

We have the following statements.

  1. If $f$ is an increasing function of $w^3$ then for the proof of $P\geq0$ it's enough to prove it for an equality case of two variables.
  1. If $f$ is a decreasing function of $w^3$ then for the proof of $P\geq0$ it's enough to prove it for an equality case of two variables.
  1. If $f$ is a concave function of $w^3$ then for the proof of $P\geq0$ it's enough to prove it for an equality case of two variables.

For non-negative variables we have the following statements.

  1. If $f$ is an increasing function of $w^3$ then for the proof of $P\geq0$ it's enough to prove it for an equality case of two variables and for $w^3=0$.
  1. If $f$ is a decreasing function of $w^3$ then for the proof of $P\geq0$ it's enough to prove it for an equality case of two variables.
  1. If $f$ is a concave function of $w^3$ then for the proof of $P\geq0$ it's enough to prove it for an equality case of two variables.
  1. If $f$ is an increasing function of $v^2$ then for the proof of $P\geq0$ it's enough to prove it for an equality case of two variables.
  1. If $f$ is a decreasing function of $v^2$ then for the proof of $P\geq0$ it's enough to prove it for an equality case of two variables.
  1. If $f$ is a concave function of $v^2$ then for the proof of $P\geq0$ it's enough to prove it for an equality case of two variables.
  1. If $f$ is an increasing function of $u$ then for the proof of $P\geq0$ it's enough to prove it for an equality case of two variables.
  1. If $f$ is a decreasing function of $u$ then for the proof of $P\geq0$ it's enough to prove it for an equality case of two variables.
  1. If $f$ is a concave function of $u$ then for the proof of $P\geq0$ it's enough to prove it for an equality case of two variables.