AM(s)-凸函数及其Jensen型不等式

摘要 针对函数的凸性及其广义凸性，研究凸函数的推广问题.首先引入了n个正数的加权r次幂s-平均的概念和记号，并利用加权r次幂s-平均定义了AM（s）-凸函数；然后用符号化的方式讨论了AM（s）-凸函数的判定定理和运算性质；最后，证明了AM（s）-凸函数的Jensen型不等式，并给出了其等价形式.研究结果表明，AM（s）-凸函数是包含众多凸函数的一类广义凸函数，运用加权r次幂s-平均定义和研究AM（s）-凸函数是对凸函数进行推广和研究的有效方法，同时也为凸函数的拓展推广和深入研究探索了一条新的途径.
Abstract：Based on the convexity and general convexity of a function, the authors study extending issues of a convex function. Firstly, the concept and sign of weighted r-th power s-mean of n positives are introduced; secondly, the AM(s)-convex function is defined by weighted r-th power s-mean; thirdly, the judgment theorem and operation properties of AM(s)-convex function are discussed; and finally, the Jensen-type inequality of the AM(s)-convex function is proved and an equivalent form is provided. The study shows that the AM(s)-convex function is a subset of general convex functions that includes many convex functions. Studying the AM(s)-convex function with the method of weighted r-th power s-mean is an effective way of extending and studying convex functions. This method explores a new approach to extending and studying convex functions.