关于Hardy型不等式的讨论  

Discussion on Hardy-type Inequalities

作　　者：周烁星 楼红卫[1] ZHOU Shuo-xing;LOU Hong-wei(School of Mathematical Sciences,Fudan University,Shanghai 200433,China)

机构地区：[1]复旦大学数学科学学院,上海200433

出　　处：《大学数学》2021年第2期114-118,共5页College Mathematics

基　　金：国家级大学生创新创业训练计划项目(202010246121);“莙政基金”大学生见习研修计划(20924)。

摘　　要：用级数的前n项的某种平均构成新级数的一般项并研究其性质,是一个非常有趣的问题.文章从一道数项级数练习题出发,联系Hardy不等式,将研究对象推广到通过一般函数获得的级数前n项的平均值.其次,类比连续型的Hardy不等式,将所得结论推广到了对[0,+∞)上可积函数在[0,x]上积分均值的研究,进一步推广了原命题.It is a very interesting problem to construct a new series with some mean value of the first n terms of a known series and study its properties.In this paper,starting from an exercise of number series and connecting with Hardy inequality,the research object is extended to the mean value obtained through general functions rather than common power mean.Secondly,by analogy with the Hardy inequality of the continuous type,the obtained conclusion can be extended to the study of the integral mean value of integrable functions on[0,+∞),which further extends the study of the original proposition.

关 键 词：级数 平均值 HARDY型不等式 

分 类 号：O178[理学—数学]