几种均值之间的比较及应用

Comparisons of Several Kinds of Mean Values and Their Applications

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.