Isoperimetric inequalities for conformal moments of plane domains

We prove a new sharp inequality for norms in weighted Bergman space. This inequality is then used to derive isoperimetric inequalities for geometric functionals which are closely related to the torsional rigidity of a simply connected domain (F. G. Avkhadiev, Sbornik: Mathematics 189 (1998), no. 12, 1739–1748).