Function Theoretic Majorant Principles

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


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.



Unit Disk Half Plane Level Line Harmonic Measure Counting Function 
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  1. 1.
    R. Nevanlinna [12].Google Scholar
  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.
    E. Lindelöf [2].Google Scholar
  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.
    Cf. O. Lehto [1].Google Scholar
  10. 2.
    F. Riesz [1].Google Scholar
  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.
    K. Löwner [1].Google Scholar
  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.
    Cf. R. Nevanlinna [12].Google Scholar
  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.
    E. Lindelöf [3].Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1970

Authors and Affiliations

  1. 1.Academy of FinlandFinland

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