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

Abstract

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. © 2010 Allerton Press, Inc.