  In this paper we study variations of the first non-trivial eigenvalues of the
two-dimensional $p$-Laplace operator, $p>2$, generated by measure preserving
quasiconformal mappings $\varphi : \mathbb D\to\Omega$, $\Omega \subset\mathbb
R^2$. This study is based on the geometric theory of composition operators on
Sobolev spaces with applications to sharp embedding theorems. By using a sharp
version of the reverse H\"older inequality we obtain lower estimates of the
first non-trivial eigenvalues for Ahlfors type domains.
