  We study the asymptotic behavior of the ratio $|f(z)|/|z|$ as $z\to 0$ for
mappings differentiable a.e. in the unit disc with non-degenerated Jacobian.
The main tools involve the length-area functionals and angular dilatations
depending on some real number $p.$ The results are applied to homeomorphic
solutions of a nonlinear Beltrami equation. The estimates are illustrated by
examples.
