Hardy-type inequalities and principle frequency of the $p-$Laplacian

We prove a sharp $L^p$ weighted Hardy inequality involving boundary distance $\delta$ for any domain $\Omega\subsetneq \mathbb R^n$. The inequality may be improved substantially under the additional assumption that $-\log \delta$ is subharmonic. Applications of these inequalities to the principle frequency of the $p-$Laplacian are given.