Necessary and sufficient conditions for symmetric homogeneous polynomial inequalities in nonnegative real variables

Let fn(x,y,z) be a symmetric homogeneous polynomial of degree n . In this paper, we give the necessary and sufficient conditions to have fn(x,y,z) 0 for n 6 and any nonnegative real numbers x,y,z . In addition, we extend some results to n = 7 and n = 8 , and then apply the proposed method to prove several elegant symmetric homogeneous polynomial inequalities of degree n , 4 n 8 . Mathematics subject classification (2010): 26D05.