| The Best Constants of Hardy Type Inequalities for $p=-1$ |
| Received:March 28, 2006 Revised:December 12, 2006 |
| Key Words:
Hardy type inequalities weight coefficient the best constant.
|
| Fund Project:the National Natural Science Foundation of China (No.10671136); the Natural Science Foundation of Sichuan Provincial Education Department (No.2005A201). |
|
| Hits: 2837 |
| Download times: 2798 |
| Abstract: |
| For $p>1$, many improved or generalized results of the well-known Hardy's inequality have been established. In this paper, by means of the weight coefficient method, we establish the following Hardy type inequality for $p=-1$: $$ \sum_{i=1}^n\left(\frac{1}{i}\sum_{j=1}^ia_j\right)^{-1}<2\sum_{i=1}^n\left(1-\frac{\pi^2-9}{3i}\right)a_i^{-1}, $$ where $a_i>0,i=1,2,\ldots,n$. For any fixed positive integer $n\geq 2$, we study the best constant $C_n$ such that the inequality $\sum_{i=1}^n\left(\frac{1}{i}\sum_{j=1}^ia_j\right)^{-1}\leq C_n\sum_{i=1}^na_i^{-1}$ holds. Moreover, by means of the Mathematica software, we give some examples. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2008.02.010 |
| View Full Text View/Add Comment |
|
|
|
