关键词: 加权r次幂s-平均, AM(s)-凸函数, 判定定理, 运算性质, Jensen型不等式

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.

Key words: weighted r-th power s-mean, AM(s)-convex function, judgment theorem, operation property, Jensen-type inequality