Estimation of the $L_p$-norms of stress functions for finitely connected plane domains

Let $u(x,G)$ be the classical stress function of a finitely connected plane domain $G$. The isoperimetric properties of the $L^p$-norms of $u(x,G)$ are studied. Payne's inequality for simply connected domains is generalized to finitely connected domains. It is proved that the $L^p$-norms of the functions $u(x,G)$ and $u^{-1}(x,G)$ strictly decrease with respect to the parameter $p$, and a sharp bound for the rate of decrease of the $L^p$-norms of these functions in terms of the corresponding $L^p$-norms of the stress function for an annulus is obtained. A new integral inequality for the $L^p$-norms of $u(x,G)$, which is an analog of the inequality obtained by F. G. Avkhadiev and the author for the $L^p$-norm of conformal radii, is proved.