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

We study the behaviour of the smallest possible constant $d(a,b,p,\epsilon )$ in Hardy inequality $${\int }_a^b{(\frac{1}{x}{\int }_a^{x}f(t)dt)}^p{x}^{\epsilon }dx\le d(a,b,p,\epsilon ){\int }_a^b[f(x){]}^p{x}^{\epsilon }dx,2\le p<\mathrm{\infty }\mathrm{.}$$ The exact rate of convergence of $d(a,b,p,\epsilon )$ is established and the ``almost extremal’’ function is found.