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浙江大学学报（理学版）
浙江大学学报（理学版）  2017, Vol. 44 Issue (4): 403-410    DOI: 10.3785/j.issn.1008-9497.2017.04.004
数学与计算机科学     
MM-凸函数及其Jensen型不等式
宋振云1, 陈少元1, 胡付高2
1. 湖北职业技术学院 机电工程学院, 湖北 孝感 432000;
2. 湖北工程学院 数学与统计学院, 湖北 孝感 432000
MM-convex function & its Jensen-type inequality
SONG Zhenyun1, CHEN Shaoyuan1, HU Fugao2
1. School of Mechanical & Electrical Engineering, Hubei Polytechnic Institute, Xiaogan 432000, Hubei Province, China;
2. School of Mathematics & Statistics, Hubei Engineering University, Xiaogan 432000, Hubei Province, China
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摘要： 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.
关键词： 凸函数;  MM-凸函数;  判定定理;  运算性质;  Jensen型不等式    
Abstract: 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.
Key words: convex function    MM-convex function    judgment theorem    operation property    Jensen-type inequality
收稿日期: 2016-09-19 出版日期: 2017-12-09
CLC:  O178.1  
基金资助: 教育部科学技术研究重点项目（212109）.
作者简介: 宋振云(1958-),ORCID:http://orcid.org/0000-0002-7373-9733,男,教授,主要从事凸分析及其应用研究,E-mail:hbsy12358@126.com.
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引用本文:

宋振云, 陈少元, 胡付高. MM-凸函数及其Jensen型不等式[J]. 浙江大学学报（理学版）, 2017, 44(4): 403-410.

SONG Zhenyun, CHEN Shaoyuan, HU Fugao. MM-convex function & its Jensen-type inequality. Journal of ZheJIang University(Science Edition), 2017, 44(4): 403-410.

链接本文:

https://www.zjujournals.com/sci/CN/10.3785/j.issn.1008-9497.2017.04.004    或    https://www.zjujournals.com/sci/CN/Y2017/V44/I4/403

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