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 iff the domain does not contain the point at infinity and has a uniformly perfect boundary. In addition, we give estimates of the constants using some known numerical characteristics of domains.