r-凸函数与几个重要不等式的联系及应用  

Relations and Applications for r-Convex Functions and Several Important Inequalities

作　　者：何祖国[1] 

机构地区：[1]四川工程职业技术学院基础部,四川德阳618000

出　　处：《四川师范大学学报（自然科学版）》2010年第5期703-706,共4页Journal of Sichuan Normal University（Natural Science）

基　　金：国家自然科学基金(10651135)资助项目

摘　　要：从凸函数定义出发,引入并研究了一类广义凸函数——r-凸函数,利用极限等数学工具,证明了它包含凸函数、拟凸函数、对数型凸函数作为其特例.同时,证明了它的一些性质,利用其性质,给出了几何平均值、代数平均值、调和平均值、柯西不等式几个重要不等式间的联系.最后,给出了r-凸函数在不等式证明中的应用.所得的结果统一和发展了一些已有的工作.In this paper,based on the definition of convex function and by using some mathematical tools such as limit,we introduce and study a new concept-r-convex function,which includes the convex function,quasi-convex function,and logarithmically convex functions as its special cases.At the same time,we prove some properties of r-convex functions.Using its nature,we give the links between several important inequalities such as geometric mean,algebraic mean,harmonic mean,Cauchy inequality etc.Finally,we give an application of the r-convex functions in proving inequalities.The results presented in this paper improve and generalize some known results.

关 键 词：凸函数 拟凸函数 对数型凸函数 R-凸函数 加权平均值 柯西不等式 

分 类 号：O178[理学—数学]