  In this paper we study integral estimates of derivatives of conformal
mappings $\varphi:\mathbb D\to\Omega$ of the unit disc $\mathbb D\subset\mathbb
C$ onto bounded domains $\Omega$ that satisfy the Ahlfors condition. These
integral estimates lead to estimates of constants in Sobolev-Poincar\'e
inequalities, and by the Rayleigh quotient we obtain spectral estimates of the
Neumann-Laplace operator in non-Lipschitz domains (quasidiscs) in terms of the
(quasi)conformal geometry of the domains. Specifically, the lower estimates of
the first non-trivial eigenvalues of the Neumann-Laplace operator in some
fractal type domains (snowflakes) were obtained.
