Abstract
Of the results in the first two chapters we shall need here above all the existence of the harmonic measure ω(z, α, G) of an are α with respect to a region G bounded by finitely many Jordan arcs (α + β) at the point z ∈ G; this measure is uniquely determined by the following conditions:
-
1.
ω(z, α, G) is harmonic and bounded in G;
-
2.
On α, ω assumes the value 1, and on the complementary are β, the value 0.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Nevanlinna [12].
We thus also take into consideration the possibility that the “region” A w may consist solely of the arcs α w .
For by possibly extending the region A w one can always achieve a situation where w tends to an interior point (of continuity) of x w : afterwards one can again let the region tend to the original A w .
F. and R. Nevanunna [1], A. Ostrowski [1].
E. Phragmén and E. Lindelöf [1]. A still sharper form of this theorem was given by F. and R. Nevanlinna [1].
For more results of this kind, cf. F. and R. Nevanlinna [1].
E. Lindelöf [2].
A schlicht region G possesses a Green’s function if and only if the set of its boundary points is not “of harmonic measure zero”. This question will be investigated more thoroughly in Chapter V.
Cf. O. Lehto [1].
F. Riesz [1].
Cf. the footnote 2 on p. 46.
This estimate is sharp, as is shown in VII, § 5.
K. Löwner [1].
G. Julia [1], C. Carathéodory [4].
The theorem was given in the above final form by Carathéodory [4], The following simple proof stems from Landau and Valiron [1]. Cf. also my note [15].
E. Landau [1], G. Schottky [1].
Cf. R. Nevanlinna [12].
To remove the exceptional nature of the point at infinity, it is best to measure distances on the sphere.
H ν (ν = 1, 2) thus consists of the set of points w to which the function value w(z) comes arbitrarily close in an arbitrary neighborhood of P on α ν . According to this definition, H ν is either a point or a continuum.
E. Lindelöf [3].
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1970 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Nevanlinna, R. (1970). Function Theoretic Majorant Principles. 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_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-85590-0_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-85592-4
Online ISBN: 978-3-642-85590-0
eBook Packages: Springer Book Archive