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 $${\int }_a^b{\left(\frac{1}{x}{\int }_a^{x}f(t)dt\right)}^{p}dx\le d(a,b){\int }_a^b[f(x){]}^{p}dx$$ and $$\sum _{k=1}^n(\frac{1}{k}\sum _{j=1}^k{a}_j{)}^p\le d_n\sum _{k=1}^n{a}_k^p.$$ The exact rate of convergence of $d(a,b)$ and $d_n$ is established and the ``almost extremal'' function and sequence are found.