Payne type inequalities for -norms of the warping functions

Let u ( x , G ) be a warping function of a multiply connected plane domain G. A new physical functional of u ( x , G ) with an isoperimetric monotonicity property is constructed. It is proved that L p - and L q -norms of the warping function satisfy sharp isoperimetric inequalities, which, besides the norms, can contain the functional u ( G ) = sup x ∈ G u ( x , G ) . As a particular case of one of these inequalities it follows the classical result of Payne for the torsional rigidity of G. Our proofs are based on the technique of estimates on level lines devised by L.E. Payne.