Twice-punctured hyperbolic sphere with a conical singularity and generalized elliptic integral

TL;DR

This paper characterizes the hyperbolic metric of a twice-punctured sphere with a conical singularity using generalized elliptic integrals, exploring its properties and applications.

Contribution

It introduces a novel representation of the hyperbolic metric via generalized elliptic integrals and analyzes its monotonicity and applications.

Findings

Explicit formula for the hyperbolic metric in terms of generalized elliptic integrals

Monotonicity properties of the metric

Applications to geometric function theory

Abstract

We describe, in terms of generalized elliptic integrals, the hyperbolic metric of the twice-punctured sphere with one conical singularity of prescribed order. We also give several monotonicity properties of the metric and a couple of applications.