Alzer不等式為
                             
對於Alzer不等式，Sandor和Ume給了一個很簡易的證明，他們的證明包含了數學歸納法的原則和其他分析的方法。而Sandor和Ume的證明中，限制r為正的實數。
在本論文中，我們將分別使用Sandor和Ume的證法，證明了Alzer不等式在任意實數r均可成立。在主要結論中，我們將任意實數r分為 r<-1；-1<r<0；r=-1；r=0，四種情況來探討。
另外，我們也提出一些Alzer不等式的推廣，有N. Elezovic 和J. Pecaric的推廣；有F. Qi和L. Debnath 的推廣。最後，我們做出類似F. Qi和L. Debnath的推廣。

The following is known as the Alzer inequality：
                             
Sandor and Ume gave a simple proof of Alzer inequality in the case that r is a positive real number，their proofs involve the principle of the mathematical induction and other analytical methods. 
We shall prove that the Alzer inequality holds for every real number r . The methods we used to prove are motivated by Sandor and Ume，respectively.
We also state some generalization of Alzer inequality given by N. Elezovic and J. Pecaric；F. Qi and L. Debnath theorem，respectively . Finally，we give a result which is similar to that of F. Qi and L. Debnath’s theorem.

目     錄

1  介紹…………………………………………………...…p1
2  Alzer不等式的簡易證明
        2.1  Sandor的證明………………….……………p1
        2.2  Ume的證明………………………….……...p2
3  主要結論
 3.1  利用Sandor方法證明Alzer不等式
      3.1.1  的情形……………………………..….....p4
      3.1.2  的情形…………………………….........p6
             3.1.3  的情形……………….…………...p7
             3.1.4  的情形……………….…….………p7
        3.2  利用Ume方法證明Alzer不等式
            3.2.1  的情形…………………….…………p8
            3.2.2  的情形…………….……….……...p9
4 推廣
        4.1  N. Elezovic 和 J. Pecaric推廣的結果…p11
        4.2  F. Qi 和 L. Debnath推廣的結果…………p12
        4.3  推廣F. Qi 和 L. Debnath的結果…………p13
參考文獻………………………………………………………p15
Contents

1  Introduction…………………………………………….p16
2  Simpler proofs of Alzer’s inequality
   2.1  Sandor’s proof………………….……………..p16
   2.2  Ume’s proof…………………………….……….p17
3  Main results
3.1  Use Sandor method to prove Alzer inequality
 3.1.1  the case  …….……..………………………...p19
 3.1.2  the case  ………..………………............p20
        3.1.3  the case  .……...…………………….p21
        3.1.4  the case  …...………………........p22
   3.2  Use Ume method to prove Alzer inequality
        3.2.1  the case  …………………...………..p22
       3.2.2  the case  ……….............……….p24
4 Generalizations
   4.1  generalization of Alzer’s inequality give by N. Elezovic and J. Pecaric.....................………p26
   4.2  generalization of Alzer’s inequality give by F. Qi and L. Debnath…...……………………………………p26
   4.3  generalization similar to that of F. Qi and 
L. Debnath……………..…………………………………..p27
References  ………………………………………..……..p30

1. H. Alzer, On an inequality of H. Minc and L. Sathre, J. Math. Appl. 179 (1993), 396-402.
2. F. Qi and L. Debnath, On a new generalization of Alzer’s inequality, J. Math. ＆ Math. Sci. Vol.23, No.12 (2000) 815-818.
3. N. Elezovic and J. Pecaric, On Alzer’s inequality, J. Math. Anal. 223 (1998) , no.1,366-369.
4. H. Minc and L. Sathre, Some inequalities involving  , Proc. Edinburgh Math. Soc. 14 (1964/65), 41-46.
5. J. Sandor , On an inequality of Alzer , J.Math.Anal. 192(1995), no. 3 , 1034-1035.
6. J. S. Ume, An elementary proof of H. Alzer’s inequality, Math.Japon.44 (1996),no.3,521-522.