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

We establish some strong sufficient conditions that the inequality \(f_4(x; y; z) \geq 0\) holds for all nonnegative real numbers \(x; y; z\), where\( f_4(x; y; z)\) is a cyclic homogeneous polynomial of degree four. In addition, in the case \(f_4(1; 1; 1) = 0\) and also in the case when the inequality \(f_4(x; y; z) \geq 0\) does not hold for all real numbers \(x; y; z\), we conjecture that the proposed sufficient conditions are also necessary that\( f_4(x; y; z) \geq 0\) for all nonnegative real numbers \(x; y; z\). Several applications are given to show the effectiveness of the proposed methods.