Function Theoretic Majorant Principles

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

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 zG; this measure is uniquely determined by the following conditions:
1. 1.

ω(z, α, G) is harmonic and bounded in G;

2. 2.

On α, ω assumes the value 1, and on the complementary are β, the value 0.

Keywords

Unit Disk Half Plane Level Line Harmonic Measure Counting Function
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References

1. 1.
2. 2.
We thus also take into consideration the possibility that the “region” A w may consist solely of the arcs α w.Google Scholar
3. 1.
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.Google Scholar
4. 1.
F. and R. Nevanunna [1], A. Ostrowski [1].Google Scholar
5. 1.
E. Phragmén and E. Lindelöf [1]. A still sharper form of this theorem was given by F. and R. Nevanlinna [1].Google Scholar
6. 1.
For more results of this kind, cf. F. and R. Nevanlinna [1].Google Scholar
7. 1.
8. 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.Google Scholar
9. 1.
10. 2.
11. 1.
Cf. the footnote 2 on p. 46.Google Scholar
12. 2.
This estimate is sharp, as is shown in VII, § 5.Google Scholar
13. 1.
14. 1.
G. Julia [1], C. Carathéodory [4].Google Scholar
15. 1.
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].Google Scholar
16. 1.
E. Landau [1], G. Schottky [1].Google Scholar
17. 1.
18. 1.
To remove the exceptional nature of the point at infinity, it is best to measure distances on the sphere.Google Scholar
19. 1.
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.Google Scholar
20. 3.