  We obtained estimates for first eigenvalues of the Dirichlet boundary value
problem for elliptic operators in divergence form (i.e. for the principal
frequency of non-homogeneous membranes) in bounded domains $\Omega \subset
\mathbb C$ satisfying quasihyperbolic boundary conditions. The suggested method
is based on the quasiconformal composition operators on Sobolev spaces and
their applications to constant estimates in the corresponding
Sobolev-Poincar\'e inequalities. We also prove a variant of the
Rayleigh-Faber-Khran inequality for a special case of these elliptic operators.
