在2000年，F. Qi 和 L. Debnath 推廣了 Alzer 不等式，而在他們的證明中限制r為正實數且{a_1,a_2,...}是一個遞增的正實數數列。 在本論文主要結論中，我們將任意正實數r設定為任意負實數且{a_1,a_2,...}是一個遞增的正實數數列更改為遞減的正實數數列，建立了一個類似的結果。

In 2000, F. Qi and L. Debnath further generalize Alzer's inequality. The main purpose of this note is to change some conditions and establish a result that similar to this inequality.

目　　　錄
1. 前言 ....................................P. 1
   定理1.1 .................................P. 1
   定理1.2 .................................P. 1
   定理1.3 .................................P. 1
   定理1.4 .................................P. 1
   定理1.5 .................................P. 2
   定理1.6 .................................P. 2
2. 主要結果 ................................P. 2
   定理2.1 .................................P. 2
   推論2.2 .................................P. 6
   註解2.3 .................................P. 7
   例題2.4 .................................P. 8
   例題2.5 .................................P. 9
   例題2.6 .................................P. 11
   例題2.7 .................................P. 12
參考文獻 ...................................P. 14

Contents
1. Introduction ............................P. 15
   Theorem 1.1 .............................P. 15
   Theorem 1.2 .............................P. 15
   Theorem 1.3 .............................P. 15
   Theorem 1.4 .............................P. 15
   Theorem 1.5 .............................P. 16
   Theorem 1.6 .............................P. 16
2. Main Results ............................P. 16
   Theorem 2.1 .............................P. 16
   Corollary 2.2 ...........................P. 20
   Remark 2.3 ..............................P. 21
   Example 2.4 .............................P. 22
   Example 2.5 .............................P. 23
   Example 2.6 .............................P. 25
   Example 2.7 .............................P. 26
References .................................P. 28

[1] H. Alzer, On an inequality of H.Minc and L. Sathre, J. Math. Anal. Appl. 179 (1993), No. 2, 396-402.
[2] N. Elezvic and J. Pecaric, On Alzer's inequality, J. Math. Anal. Appl. 223 (1998), No. 1, 366-369.
[3] J. S. Martins, Arithmetic and geometric means, an application to Lorentz sequence spaces, Math. Nachr. 139 (1988), 281-288.
[4] H.Minc and L. Sathre, Some inequalities involving (r!)^(1/r), Proc. Edinburgh Math. Soc. (2) 14 (1964-1965), 41-46.
[5] F. Qi, Generalization of H. Alzer's inequality, J. Math. Anal. Appl. 240 (1999), No. 1. 294-297.
[6] F. Qi and L. Debnath, On a new generalization of Alzer's inequality, J. Math. & Math. Sci. Vol. 23, No.12 (2000) 815-818.
[7] J. Sandor, On an inequality of Alzer, J. Math. Anal. Appl. 192 (1995), No. 3, 1034-1035.
[8] J. S. Ume, An elementary proof of H. Alzer's inequality, Math. Japon. 44 (1996), No. 3, 521-522