TITLE: How do I proof the theorem
QUESTION [6 upvotes]: 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?

REPLY [3 votes]: 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)