Relations Between Noneuclidean and Euclidean Metrics

Part of the Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen book series (GL, volume 162)


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).


Boundary Point Unit Disk Interior Point Jordan Curve Harmonic Measure 
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  1. 1.
    T. Carleman [1], A. Ostrowski [3], S. Warschawski [1].Google Scholar
  2. 1.
    Since w is bounded by hypothesis, we can assume without loss of generality that |wa| < 1 in the right half plane.Google Scholar
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    For the following, cf. T. Carleman [1].Google Scholar
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    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
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    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
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    This is, by the way, an immediate consequence of the principle of monotoneity.Google Scholar
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    One can find a sharper estimate for m w with the help of elliptic functions by mapping the rectangle conformally onto a half plane.Google Scholar
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    The above problem has been handled using other methods by Ahlfors [6]. Also cf. G. Pólya [1].Google Scholar
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    Schmidt‘s proof can be found in Carathéodory [5]. Cf. H. Grunsky [1] also.Google Scholar
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    L. Ahlfors [1].Google Scholar
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    Any continuous Jordan are whose end points are boundary points and al] of whose remaining points lie within the region is called a cross cut.Google Scholar
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    For it is at once clear that the set of accumulation points of the cross cuts (math) decomposes the region G into certain subregions a certain one of which (G′) has the point Z 1 as a boundary point and Z 2 as an exterior point. The boundary of G′ is thus cut by C at an odd number of points and consequently contains at least one cross cut that separates the points Z 1, Z2.Google Scholar
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    For an extension, cf. 4.4.Google Scholar
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Copyright information

© Springer-Verlag Berlin Heidelberg 1970

Authors and Affiliations

  1. 1.Academy of FinlandFinland

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