The results of a pair of twin symmetric inequalities are generalized,and four families of twin symmetric inequalities are obtained and proved as well.By applying the arithmetic-geometric mean inequality,the proof of the first two families of symmetric inequalities is successfully completed.However,it is no longer possible to use the arithmetic-geometric mean inequality alone when proving the latter two families of symmetric inequalities.With the help of the properties of Popoviciu's inequality and convex function,the difficulty of proof is overcome,and the conclusion of symmetric inequality between the latter two families is proved.
关键词
算术-几何平均不等式/Popoviciu不等式/对称不等式/凸函数
Key words
algorithm-geometric mean inequality/Popoviciu inequality/symmetric inequality/convex function
万方数据