| Summary: | 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.
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