Sharp Hardy type inequalities with weights depending on Bessel function

Abstract

We prove exact Hardy type inequalities with the weights depending on a Bessel function. We obtain one-dimensional $L^p$-inequalities and provide an example of extending these inequalities for the case of convex domains with a finite inner radius. The proved statements are generalization for the case of arbitrary $p\geqslant2$ of the corresponding inequality proved by F. G. Avkhadiev and K.-J. Wirths for $p=2$.