| 文家金,高朝邦.$p=-1$的Hardy型不等式的最佳常数[J].数学研究及应用,2008,28(2):316~322 |
| $p=-1$的Hardy型不等式的最佳常数 |
| The Best Constants of Hardy Type Inequalities for $p=-1$ |
| 投稿时间：2006-03-28 修订日期：2006-12-12 |
| DOI：10.3770/j.issn:1000-341X.2008.02.010 |
| 中文关键词: Hardy型不等式 权系数 最佳常数. |
| 英文关键词:Hardy type inequalities weight coefficient the best constant. |
| 基金项目:国家自然科学基金(No.10671136); 四川省教育厅科学基金(No.2005A201) |
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| 中文摘要: |
| 著名的Hardy不等式在$p>1$的情形已有一些改进或推广.本文采用权系数方法对$p=-1$的情形建立如下具有最佳系数的Hardy型不等式：$$\sum_{i=1}^n\left(\frac{1}{i}\sum_{j=1}^ia_j\right)^{-1}<2\sum_{i=1}^n\left(1-\frac{\pi^2-9}{3i}\right)a_i^{-1}, \quad a_i>0,i=1,2,\cdots,n;$$对于固定的正整数$n\geq 2$，研究了使不等式$\sum_{i=1}^n\le |
| 英文摘要: |
| 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. |
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