  This paper is devoted to Hardy type inequalities with remainders for
compactly supported smooth functions on open sets in the Euclidean space. We
establish new inequalities with weight functions depending on the distance
function to the boundary of the domain. One-dimensional $L_1$ and $L_p$
inequalities and their multidimensional analogues are proved. We consider
spatial inequalities in open convex domains with the finite inner radius.
Constants in these inequalities depend on the roots of parametric Lamb equation
for the Bessel function and turn out to be sharp in some particular cases.
