Rellich type inequalities for polyharmonic operators in plane domains

For polyharmonic operators we study Rellich type inequalities in plane domains. We prove that these inequalities are pithy if and only if the boundary of the domain is a uniformly perfect set. We prove that the constants in these inequalities are determined exactly for convex domains as well as for domains that are close to convex in certain sense. Also, it is investigated new versions of the main inequality, when weight functions are powers of the coefficient of the Poincaré metric.