Upper estimates for hyperbolic metrics on subdomains: the case of equality

Abstract

It is the purpose of this paper to supplement our previous work concerning upper estimates for hyperbolic metrics on subdomains. Specifically, our interest is in determining when equality holds in the relevant inequality to clarify the situation that has failed to be explicitly mentioned before. The method of proof again involves a standard technique based on the classical Schwarz–Pick lemma.

Keywords:

• hyperbolic metric
• upper estimate
• subdomain
• Schwarz–Pick lemma
• Fuchsian group

AMS Subject Classifications:

• Primary: 30F45
• 30C80
• Secondary: 30C25
• 30F35

Acknowledgements

The author is pleased to thank the referee(s) for careful readings of the manuscript and for helpful suggestions.

Notes

No potential conflict of interest was reported by the authors.

1 We have made use only of the equality (2.4(2.4) ), which could be easily derived from (4.5(4.5) ).