Quasiconformal and HQC mappings between Lyapunov Jordan domains

Let h be a quasiconformal (qc) mapping of the unit disk U onto a Lyapunov domain. We show that h maps subdomains of Lyapunov type of U, which touch the boundary of U, onto domains of similar type. In particular if h is a harmonic qc (hqc) mapping of U onto a Lyapunov domain, using it, we prove that h is co-Lipschitz (co-Lip) on U. This settles an open intriguing problem.