On $\mu$-conformal homeomorphisms and boundary correspondence
  We study the boundary correspondence under $\mu$-homeomorphisms $f$ of the open upper half-plane onto itself. Sufficient conditions are given for $f$ to admit a homeomorphic extension to the closed half-plane with prescribed boundary regularity. The proofs are based on the modulus estimates for semiannuli in terms of directional dilatations of $f$ which might be of independent interest. 
Vladimir Gutlyanskii, Ken-ichi Sakan and Toshiyuki Sugawa
math.CV