Hardy's inequalities with remainders and lamb-type equations

We study Hardy-type integral inequalities with remainder terms for smooth compactly-supported functions in convex domains of finite inner radius. New $L_1$- and $L_p$-inequalities are obtained with constants depending on the Lamb constant which is the first positive solution to the special equation for the Bessel function. In some particular cases the constants are sharp. We obtain one-dimensional inequalities and their multidimensional analogs. The weight functions in the spatial inequalities contain powers of the distance to the boundary of the domain. We also prove that some function depending on the Bessel function is monotone decreasing. This property is essentially used in the proof of the one-dimensional inequalities. The new inequalities extend those by Avkhadiev and Wirths for $p= 2$ to the case of every $p \geq 1$.