The Best Constants of Hardy Type Inequalities for $p=-1$

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.