MM-凸函数及其Jensen型不等式

考虑函数的广义凸性问题,利用区间上的二元幂平均定义了MM-凸函数,讨论了MM-凸函数的若干判定定理及运算性质,建立了其Jensen型不等式,并给出了Jensen型不等式的等价形式及推论.结果表明,MM-凸函数是比较函数定义区间内任意两点的幂平均函数值与其函数值的幂平均大小所确定的各类凸函数的推广.MM-凸函数概念的引入,为深入研究凸函数和拓展凸函数概念探索了一条新途径.

MM-convex function & its Jensen-type inequality

Considering the general convexity of functions, the authors present the definition of MM-convex function with two variables power means within the interval. Based on the definition, this article discusses its judgment theorems and operation properties, sets up its Jensen-type inequality, and provides the equivalent form of Jensen-type inequality and the deduction. Results show that MM-convex function is an extension of all convex functions determined by the power mean value of two arbitrary points within the definition domain of comparison function and by the power mean of the value. The introduction of MM-convex function brings an effective approach to deep study and further extension of convex function.