Carath\'eodory density of the Hurwitz metric on plane domains

TL;DR

This paper introduces a new Carathéodory-type metric that generalizes the Hurwitz metric on plane domains and explores its fundamental properties, extending the understanding of hyperbolic metrics in complex analysis.

Contribution

The paper defines a novel metric extending the Hurwitz metric in the Carathéodory framework and investigates its basic properties and connections.

Findings

The new metric generalizes the Hurwitz metric.

It retains key properties similar to the hyperbolic metric.

Connections between the new metric and classical metrics are established.

Abstract

It is well-known that the Carath\'eodory metric is a natural generalization of the Poincar\'e metric, namely, the hyperbolic metric of the unit disk. 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 hyperbolic metric when the domains are simply connected. In this paper, we define a new metric which generalizes the Hurwitz metric in the sense of Carath\'eodory. Our main focus is to study its various basic properties in connection with the Hurwitz metric.