some strong sufficient conditions for cyclic homogeneous polynomial inequalities of degree four in real variables

We establish some strong sufficient conditions such that the inequality $f_4(x,y,z)ge 0$ holds for any real numbers $x,y,z,$ where $f_4(x,y,z)$ is a cyclic homogeneous polynomial of degree four. In addition, we give a conjecture which states some new necessary and sufficient conditions such that the inequality $f_4(x,y,z)ge 0$ holds for any real numbers $x,y,z.$ Several applications are given to show the effectiveness of the proposed method.