Journal of Inequalities and Applications
Volume 6 (2000), Issue 3, Pages 253-260
doi:10.1155/S1025583401000157
Isoperimetric inequality fortorsional rigidity in the complex plane
Chebotarev Institute of Mathematics and Mechanics, Kazan State University, Universitetskaya 17, Kazan 420008, Russia
Received 6 August 1999; Revised 28 September 1999
Copyright © 2000 R. G. Salahudinov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Suppose SZ is a simply connected domain in the complex plane. In (F.G. Avhadiev, Matem. Sborn., 189(12) (1998), 3–12 (Russian)), Avhadiev introduced new geometrical functionals, which give two-sided estimates for the torsional rigidity of Ω. In this paper we find sharp lower bounds for the ratio of the torsional rigidity to the new functionals. In particular, we prove that
3Ic(∂Ω)≤2P(Ω),
where P(Ω) is the torsional rigidity of Ω,
Ic(∂Ω)=∫∫ΩR2(z,Ω)dxdy
and R(z,Ω) is the conformal radius of Ω at a point z.