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This is a very useful method for the proof of polynomial inequalities with three variables. Sometimes it works for more variables.

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.

  2. 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.

  3. 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 $w^3=0$.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. 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.

  8. 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.

  9. 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.

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