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).
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References
T. Carleman [1], A. Ostrowski [3], S. Warschawski [1].
Since w is bounded by hypothesis, we can assume without loss of generality that |w — a| < 1 in the right half plane.
For the following, cf. T. Carleman [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.
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.
This is, by the way, an immediate consequence of the principle of monotoneity.
One can find a sharper estimate for m w with the help of elliptic functions by mapping the rectangle conformally onto a half plane.
The above problem has been handled using other methods by Ahlfors [6]. Also cf. G. Pólya [1].
P. Koebe [1].
Schmidt‘s proof can be found in Carathéodory [5]. Cf. H. Grunsky [1] also.
G. Pick [2], R. Nevanlinna [1].
L. Ahlfors [1].
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.
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.
For an extension, cf. 4.4.
A. Beurling [1], R. Nevanlinna [11].
In addition to the above named work of Beurling and of the author, cf. E. Landau [3], W. Fenchel [1], E, Schmidt [1].
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© 1970 Springer-Verlag Berlin Heidelberg
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Nevanlinna, R. (1970). Relations Between Noneuclidean and Euclidean Metrics. In: Eckmann, B., van der Waerden, B.L. (eds) Analytic Functions. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, vol 162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-85590-0_5
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DOI: https://doi.org/10.1007/978-3-642-85590-0_5
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