How do I proof the theorem

Question

Theorem:

Let $f(x,y,z)$ be a cyclic polynomial of degree $3$.

The inequality $f(x,y,z) \ge 0$ holds for all non negative variables $x,y,z$ if and only if:

$f(x,x,x)\ge0$,

$f(x,y,0) \ge 0\ ,\ \forall (x,y)\ge 0 $

How do I start the proof of the theorem?

The homogeneous version is Stolarsky's cyclic $3$ variable inequality. However I can't find any useful links to Stolarsky's Inequality by googling (strange). — r9m, Jan 08 '15 at 16:01
Search for "CD3-improved" by Pham Kim Hung, sample available at http://gil.ro/downloadable/download/sample/sample_id/1/ — Macavity, Jan 08 '15 at 16:05
@Macavity You might want to put this reference as an answer. — Wojowu, Jan 08 '15 at 17:08
@Wojowu Lets see if someone gets original or has a better reference... — Macavity, Jan 08 '15 at 17:10

score 3 · Answer 1 · answered Jan 09 '15 at 04:36

Search for the "CD3-improved" theorem by Pham Kim Hung, there is an excerpt / sample available.

The corresponding theorem for third degree symmetric polynomials is attributed to Hoo Joo Lee, the cyclic case is of course a generalisation.

(It would certainly be interesting to see a simpler proof)

1 Answers1