Geodesic
Isoperimetric monotony of the $L^p$-norm of the warping function of a~plane simply connected domain
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2010), pp. 59-68

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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.
Keywords: torsional rigidity, isoperimetric inequalities, isoperimetric monotony, Schwarz symmetrization, Kohler-Jobin symmetrization. 
@article{IVM_2010_8_a6,
     author = {R. G. Salakhudinov},
     title = {Isoperimetric monotony of the $L^p$-norm of the warping function of a~plane simply connected domain},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {59--68},
     publisher = {mathdoc},
     number = {8},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2010_8_a6/}
}
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R. G. Salakhudinov. Isoperimetric monotony of the $L^p$-norm of the warping function of a~plane simply connected domain. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2010), pp. 59-68. http://geodesic.mathdoc.fr/item/IVM_2010_8_a6/