A generalized Hurwitz metric

TL;DR

This paper introduces a generalized Hurwitz metric extending the original definition to arbitrary subdomains of the complex plane, exploring its properties and relation to the classical Poincaré hyperbolic metric.

Contribution

It provides an alternative definition of the Hurwitz metric and develops a generalized version applicable to all subdomains, analyzing their properties and intersections.

Findings

The generalized Hurwitz metric is well-defined on arbitrary subdomains.

The paper identifies conditions under which the Hurwitz and hyperbolic metrics coincide.

Properties of the generalized Hurwitz metric are systematically analyzed.

Abstract

In 2016, the Hurwitz metric was introduced by D. Minda in arbitrary proper subdomains of the complex plane and he proved that this metric coincides with the Poincar\'e's hyperbolic metric when the domains are simply connected. In this paper, we provide an alternate definition of the Hurwitz metric through which we could define a generalized Hurwitz metric in arbitrary subdomains of the complex plane. This paper mainly highlights various important properties of the Hurwitz metric and the generalized metric including the situations where they coincide with each other.