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Some Strong Sufficient Conditions for Cyclic Homogeneous Polynomial Inequalities of Degree Four in Real VariablesKeywords: Cyclic homogeneous inequality , fourth degree polynomial , three real variables , strong sufficient conditions , necessary conditions Abstract: 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.
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