几种均值之间的比较及应用  

Comparisons of Several Kinds of Mean Values and Their Applications

作　　者：王桂云[1] WANG Gui-yun(Mathematics Teaching and Research Group,Zhejiang Institute of Communications,Hangzhou 311112,China)

机构地区：[1]浙江交通职业技术学院数学教研组,浙江杭州311112

出　　处：《数学的实践与认识》2022年第12期259-266,共8页Mathematics in Practice and Theory

摘　　要：设a和b为两个正实数,对基本不等式2/(1/a+1/b)≤√ab≤a+b/2≤((a^(2)+b^(2))/2)^(1/2)(当且仅当a=b时取等号)作各种加权平均、r-幂离散型及连续型平均的推广,并给出其r-幂平均关于r的单调非减性证明.特别地,分别给出了调和平均、积分平均在实际问题和重要不等式中的应用.Let a and b be two positive real numbers.We extend the basic inequality:2/(1/a+1/b)≤√ab≤a+b/2≤((a^(2)+b^(2))/2)^(1/2)(by taking"="if and only if a=b)to the weighted averages,and the r-power averages for discrete and continuous settings.The increasing monotonicity of the r-th power average for discrete and continuous cases with respect to the independent variable r is proved,respectively.In particular,the applications of harmonic average and integral average to various practical problems and important inequalities are presented,respectively.

关 键 词：调和平均 离散及连续型的r-幂平均 加权平均 r-幂平均的单调性 

分 类 号：O178[理学—数学]