A four-point criterion for the Möbius property of a homeomorphism of plane domains

An ordered quadruple of pairwise distinct points $T=\{z_1,z_2,z_3,z_4\}\subset\mathbf C$ is called regular whenever $z_2$ and $z_4$ lie at the opposite sides of the line through $z_1$ and $z_3$. Consider $\Phi(T)=\angle z_1z_2z_3+\angle z_1z_4z_3$ (the angles are undirected) as some geometric characteristic of a regular tetrad. We prove the following theorem: For every fixed $\alpha\in(0,2\pi)$ the Möbius property of a homeomorphism $f\colon D\to D^*$ of domains in $\mathbf C$ is equivalent to the requirement that each regular tetrad $T\subset D$ with $\Phi(T)=\alpha$ whose image $fT$ is also a regular tetrad satisfies $\Phi(fT)=\alpha$. In 1994 Haruki and Rassias established this criterion for the Möbius property only in the class of univalent analytic functions $f(z)$.