Optimal power-weighted Hardy inequalities on finite intervals

We extend a recently derived optimal Hardy inequality in integral form on finite intervals by Dimitrov, Gadjev, and Ismail \cite{DGI24} to the case of additional power weights and then derive an optimal power-weighted Hardy inequality in differential form on finite intervals, noting that the optimal constant of the latter inequality differs from the former. We also derive an optimal multi-dimensional version of the power-weighted Hardy inequality in differential form on spherical shell domains.