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

Quasiconformal and HQC mappings between Lyapunov Jordan domains