On the Constants and Extremal Function and Sequence for Hardy Inequalities in $L_p$ and $l_p$

Abstract

We study the behavior of the smallest possible constants $d(a,b)$ and $d_n$ in Hardy inequalities $$ \int_a^b\left(\frac{1}{x}\int_a^xf(t)dt\right)^p\,dx\leq d(a,b)\,\int_a^b [f(x)]^p dx $$ and $$ \sum_{k=1}^{n}\Big(\frac{1}{k}\sum_{j=1}^{k}a_j\Big)^p\leq d_n\,\sum_{k=1}^{n}a_k^p. $$ The exact rate of convergence of $d(a,b)$ and $d_n$ is established and the ``almost extremal'' function and sequence are found.