Some strong sufficient conditions for cyclic homogeneous polynomial inequalities of degree four in nonnegative variables

We establish some strong sufficient conditions that the inequality f4(x;y;z)≥0 holds for all nonnegative real numbers x;y;z, where f4(x;y;z) is a cyclic omogeneous polynomial of degree four. In addition, in the case f 4(1;1;1)=0 and also in the case when the inequality f4(x;y;z)≥0 does not hold for all real numbers x;y;z, we conjecture that the proposed sufficient conditions are also necessary that f4(x;y;z)≥0 for all nonnegative real numbers x;y;z. Several applications are given to show the effectiveness of the proposed methods.