An analog of the Poincaré metric and isoperimetric constants

For plane domains we define a new metric close to the Poincaré metric with the Gaussian curvature $k=-4$. For this quasi-hyperbolic metric we study inequalities of isoperimetric type. It is proved that the constant of the linear quasi-hyperbolic isoperimetric inequality for admissible subdomains of a given domain is finite if and only if the domain does not contain the point at infinity and has a uniformly perfect boundary. Also, we give estimates of these constants using some known numerical characteristics of domains.