Isoperimetric monotony of the $L^p$-norm of the warping function of a plane simply connected domain

Let $G$ be a simply connected domain and let $u(x,G)$ be its warping function. We prove that $L^p$-norms of functions $u$ and $u^{-1}$ are monotone with respect to the parameter $p$. This monotony also gives isoperimetric inequalities for norms that correspond to different values of the parameter $p$. The main result of this paper is a generalization of classical isoperimetric inequalities of St. Venant–Pólya and the Payne inequalities.