臺灣博碩士論文加值系統

在第一篇論文中，Alzer 證明了下列結果：
若r > 0 且n■ N，則
■.
我們在 ■ 和■ 之間插入一項 得到下列之不等式：
定理：若 r 是一個正實數且n是一個正整數，則
■.
同理，由 Kuang 的下列不等式：
若 f 在 [0,1] 區間是一個非線性的凹函數，則
■.
我們在 ■ 的左邊加入以下的不等式：
定理：若 f 在 [0,1] 區間是一個嚴格遞增的凹(凸)函數，則對每一個非負整數 k ,數列 ■ 將隨者 n 的增加而增加並且有一個上界 ,即
■,
這裡的 n 是一個自然數.

In [1],Alzer proved the following inequality：
If r is a positive real number and if n is a positive integer, then
■ .
We insert between ■ and ■ a term and get
Theorem：If r is a positive real number and if n is a positive integer, then
■.
Similarly, for the inequality by Kuang in [2].
If f is a nonlinear convex function on [0,1],then
■.
We develop an inequality on the left hand side of■and get
Theorem：Let f be a strictly increasing convex (or concave) function on [0,1].Then for each nonnegative integer k the sequence ■ is increasing in n and has an upper bound■ ,that is
■
Where n is a natural number.
二.主要部分.........................................3~13
四.參考文獻.........................................14

1.H.Alzer, On inequality of H. Minc and L. Sathre, J. Math. Anal. 179(1993), 396-402.
2.Ji-chang Kuang, some extensions and refinements of Min-Sathre inequality, Math. Gaz. 83(1999),123-127.
3.D.S. Mitrinorbic, J.E. Pecaric and A.M. Fink, Classocal and New inequalities in Analysis, Kluwer Academic Publishers, Dordrecht/Boston/London, 19993.
4.Feng Qi, Generations of Alzer's and Kuang's inequality. Tamkang J. Math. Vol 31. No.3(to appear).