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) > =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) > =0 holds for any real numbers x,y,z Several applications are given to show the effectiveness of the proposed method.