假設x及y是不同的正數，則 
Mp = Mp (x , y) 
稱冪平均數， 
L = L (x , y) 
稱對數平均數。
 
Chang[2]已證明 : 
在某些限制下的x和y，當q<1/3 時，L < Mq，並且證明當 p>0，Mp < L 。
 
我們將證明在某些限制下的x和y，當q <1/6 時，則L < Mq，並且證明當p > 0，L > Mp。

If x , y are distinct positive numbers, then 
Mp= Mp (x , y) 
is called the power mean of x and y . 
L = L (x , y) 
is called the Logarithmic mean of x and y . 

Chang[2]has proved that : 
If L < Mq holds for some q such that q<1/3 ，as well as   Mp < L holds for some p such that p>0 under certain restrictions on  x and y 。. 

We shall prove that L < Mq holds for some q such that q <1/6 , as well as L > Mp holds for some p such that p > 0 under certain restrictions on x and y .

1.前言……………………………………………………………p.1
2.主要結果………………………………………………………p.3
3.參考文獻……………………………………………………p.25
圖表目錄
圖1…………………………………………………………………p.4
圖2…………………………………………………………………p.4
圖3…………………………………………………………………p.8
圖4…………………………………………………………………p.8
圖5…………………………………………………………………p.12
圖6…………………………………………………………………p.13
圖7…………………………………………………………………p.17
圖8…………………………………………………………………p.17
圖9…………………………………………………………………p.21
圖10………………………………………………………………p.22
1.Introduction……………………………………………………p.26
2.Main results………………………………………………… p.28 
3.References……………………………………………………p.48 
List of figures
Figure 1……………………………………………………………p.28
Figure 2……………………………………………………………p.29
Figure 3……………………………………………………………p.32
Figure 4……………………………………………………………p.33
Figure 5……………………………………………………………p.36
Figure 6……………………………………………………………p.37
Figure 7……………………………………………………………p.40
Figure 8……………………………………………………………p.41
Figure 9……………………………………………………………p.44
Figure 10……………………………………………………………p.45

1.B.C.Carlson , The logarithmic mean , Amer. Math. Monthly,79(1972) 615-618

2.Bi-Ching Chang, On relation between the power mean and the Logarithmic mean ,Master Thesis , Department of Mathematics , Tamkang University.(2005).

3.N.D.Kazarinoff , Analytic Inequalities , Holt, New York 1961.

4.T.P.Lin, The Power Mean and The Logarithmic Mean, Amer. Math. Monthly,81(1974)879-883

5.D.S.Mitrinovic, Analytic Inequalities,Spring-Verlag,Berlin,1970, pp.272,237(section 3.6.15 and 3.6.17)