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

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.