M\"obius metric in sector domains

The M\"obius metric $\delta_G$ is studied in the cases where its domain $G$ is an open sector of the complex plane. We introduce upper and lower bounds for this metric in terms of the hyperbolic metric and the angle of the sector, and then use these results to find bounds for the distortion of the M\"obius metric under quasiregular mappings defined in sector domains. Furthermore, we numerically study the M\"obius metric and its connection to the hyperbolic metric in polygon domains.