四族姊妹对称不等式的证明  

The Proof of Four Families of Twin Symmetric Inequalities

作　　者：叶瑞松[1] YE Ruisong(College of Science,Shantou University,Shantou 515063,China)

机构地区：[1]汕头大学理学院,广东汕头515063

出　　处：《徐州工程学院学报（自然科学版）》2024年第2期1-8,共8页Journal of Xuzhou Institute of Technology(Natural Sciences Edition)

基　　金：国家自然科学基金项目(11771265);广东省基础与应用基础研究基金项目(2023A1515030199)。

摘　　要：推广一对姊妹对称不等式的结果,得到四族姊妹对称不等式,并加以证明.应用算术-几何平均不等式便可顺利完成对前两族对称不等式的证明.在证明后两族对称不等式时,单独使用算术-几何平均不等式已经无法完成.通过借助Popoviciu不等式和凸函数的性质,克服证明碰到的困难,使得后两族对称不等式的结论得到证明.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 

分 类 号：O178[理学—数学]