A note on the logarithmic mean

• Main Authors: Xu, Hong-Quan, 徐宏全
• Other Authors: Gou-Sheng Yang
• Format: Others
• Language: zh-TW
• Published: 1997
• Online Access: http://ndltd.ncl.edu.tw/handle/53151921314114529654

碩士 === 淡江大學 === 數學學系 === 85 === For unequal positive x and y, the arithmetic mean A(x,y), the identric mean I(x,y), the logarithmic mean L(x,y), the geometric mean G(x,y), and the harmonic mean H(x,y) for x and y are defined by A=A(x,y)=(x+y)/2, I=I(x,y)=(1/e)(x^x/y^y) ^[1/(x-y)] L=L(x,y)=(x-y)/(lnx-lny), G=G(x,y)=(xy)^(1/2) H=H( x,y)=(2xy)/(x+y) respectively. In 1972, B.C.Carlson proved that G<L<(2G+A)/3<A. In 1975, K.B.Stolarsky proved that G<L<I<A. In 1990, J. Sandor proved that (A+L)/2<I. In 1991, J. Sandor proved that (A+L)/2<(2A+G)/3<I. Consequently, we have y<H<G<L<(2G+A)/3<(A+L)/2<(2A+G)/3<I<A<x if x>y. In this article, we present an elementary proof for the inequalities. The idea is that, we consider the strictly increasing function F(s)=s(x-y)+y, s is between 0 and 1, and let F(s1)=H, F(s2)= G, F(s3)=L, F(s4)=(2G+A)/3, F(s5)=(A+L)/2, F(s6)=(2A+G)/3. F( s7)=I, F(s8)=A. The main results of this article is to show that 0<s1<s2<s3< s4<s5<s6<s7<s8<1,then F(0)<F(s1)<F(s2)<F(s3)<F(s4)<F(s5)<F( s6)<F(s7)<F(s8)<1, which is equivalent to the inequalities.