Analytic Functions pp 67-113 | Cite as

# Relations Between Noneuclidean and Euclidean Metrics

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## Abstract

The usefulness of various noneuclidean metrics (harmonic measure, hyperbolic measure) in function theory is due in part to these metrics being *conformai invariants*, the designation for any quantity that behaves invariantly relative to the group of conformai mappings. But in addition, it turns out that for a variety of questions it is just by applying concepts from noneuclidean geometry that certain features can be sharply delineated; this is often true for the characterization of various extremal properties, for example. The introduction of such metrics is therefore altogether natural, and we would seem to be justified in developing the theory of such metrics systematically, not worrying about their relationship to the usual metrics (euclidean or spherical).

## Keywords

Boundary Point Unit Disk Interior Point Jordan Curve Harmonic Measure## Preview

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## References

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- 1.Since
*w*is bounded by hypothesis, we can assume without loss of generality that |*w*—*a*| < 1 in the right half plane.Google Scholar - 1.For the following, cf. T. Carleman [1].Google Scholar
- 1.The reader is invited to prove this theorem directly as a consequence of the minimum principle, considering the harmonic measure log |
*w*| and using the harmonic minorant constructed above. This direct proof has the advantage of showing that the theorem holds without any kind of restrictive assumptions about the region’s boundary.Google Scholar - 1.
*D*_{σ}may consist of several components. If*t*is a point of the region, then one always understands*D*_{σ}to be that component containing the point*t*.Google Scholar - 2.This is, by the way, an immediate consequence of the principle of monotoneity.Google Scholar
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