关于算术平均和几何平均的加权算术平均和加权几何平均

The Weighted Arithmetic and Geometric Means ofthe Arithmetric Mean and the Geometric Mean

设Ｇ（ａ，ｂ），Ａ（ａ，ｂ）和Ｌ（ａ，ｂ）分别是两不相等正数ａ和ｂ的几何平均、算术平均和对数平均．本文得以下两重要结论①Ｌ＜ｐＧ＋（１－ｐ）Ａ成立的充要条件为－∞＜ｐ≤２３；②ＧｐＡ１－ｐ＜Ｌ成立的充要条件为２３≤ｐ＜＋∞．

Abstract Assume a and b to be two different positive number, let G(a,b), A(a,b) and L(a,b) be the unweighted geometric, arithmetic and logarithmic means of a and b, then two inequalities were obtained as follows:① L<pG+(1-p)A holds when and only when -∞<p≤23.② GpA1-p<L holds when and only when 23≤p<+∞.