Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2020; 57(5): 1127-1142

Online first article July 10, 2020      Printed September 30, 2020

https://doi.org/10.4134/BKMS.b190789

Copyright © The Korean Mathematical Society.

A generalized Hurwitz metric

Arstu, Swadesh Kumar Sahoo

Simrol, Khandwa Road; Simrol, Khandwa Road

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.

Keywords: Hyperbolic metric, Kobayashi metric, Hurwitz metric, Hurwitz covering, generalized Hurwitz metric, hyperbolic domain, Lipschitz domain

MSC numbers: Primary 30F45; Secondary 30C20, 30C80

Supported by: The research work of Arstu was supported by CSIR-UGC $($Grant No: 21/06/2015(i)EU-V$)$ and of S. K. Sahoo was supported by NBHM, DAE $($Grant No: $2/48 (12)/2016/${\rm NBHM (R.P.)/R \& D II}/$13613)$