On the Constant and Extremal Function for Weighted Hardy Inequality in $L_p$

Abstract

We study the behaviour of the smallest possible constant $d(a,b, p,ε)$ in Hardy inequality $$ \int_a^b\left(\frac{1}{x}\int_a^xf(t)dt\right)^px^ε\,dx\leq d(a,b,p,ε)\,\int_a^b [f(x)]^px^ε\, dx, \quad 2\le p<\infty. $$ The exact rate of convergence of $d(a,b,p,ε)$ is established and the ``almost extremal&#39;&#39; function is found.