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

If x , y are distinct positive numbers, then 
         M = 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 .
Lin.[3] has proved that :
If Mp < L then p 0, and if L < Mq then q    .
We shall prove that L < Mq holds for some q such that q <  , as well as L > Mp holds for some p such that p > 0 under certain restrictions on x and y .

1. 中文前言………………………………………………… p.1~p.2
2. 中文主要結果 ………………………………………….p.2~p.13
3. 中文參考文獻………………………………………………. p.13
4. 英文前言………………………………………………… p.14~p.15
5. 英文主要結果 ………………………………………….p.15~p.27
6. 英文參考文獻………………………………………………. p.27

1.	B. C. Carlson ,The logarithmic mean, Amer. Math. Monthly,79 (1972) 615- 618 .
2.	N. D. Kazarinoff , Analytic Inequalities, Holt, New York 1961.
3.	T. P. Lin, The Power Mean and The Logarithmic Mean, Amer. Math. Monthly, 81(1974)879-883. 
4.	D. S. Mitrinovic , Analytic Inequalities, Spring-Verlag, Berlin, 1970, pp.272, 237 (sections 3.6.15 and 3.6.17)