Integral Properties of the Classical Warping Function of a Simply Connected Domain

Let $u(z,G)$ be the classical warping function of a simply connected domain $G$. We prove that the $L^p$-norms of the warping function with different exponents are related by a sharp isoperimetric inequality, including the functional $u(G)=\sup_{x\in G}u(x,G)$. A particular case of our result is the classical Payne inequality for the torsional rigidity of a domain. On the basis of the warping function, we construct a new physical functional possessing the isoperimetric monotonicity property. For a class of integrals depending on the warping function, we also obtain a priori estimates in terms of the $L^p$-norms of the warping function as well as the functional $u(G)$. In the proof, we use the estimation technique on level lines proposed by Payne.