HARMONIC DOUBLING CONDITION AND JOHN DISKS

A Jordan domain D in C is said to be a c-quasidisk if there exists a constant $c \geq 1$ such that each two points $z_1$ and $z_2$ in D can be joined by an arc $\tau$ in D such that $$ \ell(\tau) \leq c$\mid$z_1 - z_2$\mid$ $$ and $$ (1.1) min(\ell(\tau_1),\ell(\tau_2)) \leq c d(z, \partial D) $$ for all $z \in \tau$, where $\tau_1$ and $\tau_2$ are the components of $\tau\{z}$. Quasidisks have been extensively studied and can be characterized in many different ways [1],[2],[3].