Abstract
Three hyperbolic-type metrics including the triangular ratio metric, the j^*-metric, and the Möbius metric are studied in an annular ring. The Euclidean midpoint rotation is introduced as a method to create upper and lower bounds for these metrics, and their sharp inequalities are found. A new Möbius-invariant lower bound is proved for the conformal capacity of a general ring domain by using a symmetric quantity defined with the Möbius metric.
Highlights
Given two points x, y in a domain G, their intrinsic distance indicates how these points are located with respect to both each other and the domain’s boundary ∂G
One way to measure these kinds of distances is the hyperbolic metric, but there are numerous other hyperbolic-type and intrinsic metrics that can be used
We focus on three different metrics to study the intrinsic geometry of the annular ring R(r, 1) = {z ∈ C | r < |z| < 1} with 0 < r < 1
Summary
Y in a domain G, their intrinsic distance indicates how these points are located with respect to both each other and the domain’s boundary ∂G. One way to measure these kinds of distances is the hyperbolic metric, but there are numerous other hyperbolic-type and intrinsic metrics that can be used. We focus on three different metrics to study the intrinsic geometry of the annular ring R(r , 1) = {z ∈ C | r < |z| < 1} with 0 < r < 1. Since there is no explicit formula for this infimum, the hyperbolic metric is not very well suited for measuring the intrinsic distances in this kind of domain. 4, we introduce Theorem 4.1 that can be used to compute the Möbius metric in an annular ring, and present sharp inequalities between the three hyperbolic-type metrics considered.
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