Functions of Bounded Type

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


Since the characteristic function T(r) of a function meromorphic for \( \left| z \right| < R\underline{\underline < } \infty \) increases with r, the limit
$$ T\left( R \right) = \mathop {\lim }\limits_{r = R} T\left( r \right) $$
certainly exists, and there are two cases to be considered, according as T(R) = ∞ or T(R) < ∞.


Unit Disk Harmonic Measure Blaschke Product Bounded Type Radial Limit 
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  1. 1.
    H. Poincaré [2], W. Blaschke [1].Google Scholar
  2. 2.
    R. Nevanlinna [3].Google Scholar
  3. 1.
    Nonconstant, continuous functions ψ of bounded variation with a derivative that vanishes almost everywhere actually exist. A simple example of such a function can be constructed in the following way using Cantor‘s technique for the construction of perfect nowhere dense point sets (V, § 6). Distribute a positive mass h n(t) on the interval (0, t) so that one covers all 2” intervals Δ n of the approximating set E(pn) (p > 2) with a homogeneous mass of magnitude 2−n. The limit function h(t) = lim h n is a monotonic, continuous function of t that has variation 1 and whose derivative h′ vanishes for all t except for the points in the Cantor null set E(p∞).Google Scholar
  4. 1.
    P. Fatou [1].Google Scholar
  5. 2.
    Cf. e.g. H. Lebesgue [1].Google Scholar
  6. 1.
    For every measurable function u(ϑ) the integral (math) exists, where M is finite. As M → ∞, this integral tends to a limit that is either finite or infinite.Google Scholar
  7. 1.
    This holds even for nontangential approach, as can be deduced from theorems which follow.Google Scholar
  8. 1.
    This theorem shows that the relation T(r) = O(log r) is not only a necessary but also a sufficient condition for a meromorphic function w to be rational (cf. VI, 2,5). For then it follows that (2.1) is satisfied with q = 0. Since according to the first main theorem N(r, 0) and N(r, ∞) are likewise O (log r), the number of zeros and poles is finite, and the expression on the right in (2.2) is a rational function.Google Scholar
  9. 1.
    Cf. E. Borel [1], G. Valiron [3].Google Scholar
  10. 1.
    G. Valiron [1].Google Scholar
  11. 1.
    As is customary, [a] stands for the largest integer ≦ a.Google Scholar
  12. 1.
    The value α = − 1 would require special treatment, and for the sake of brevity we exclude it from consideration.Google Scholar
  13. 1.
    One has to first apply the theorem in the subregion of D 1 lying inside of |t| = R > r 1; then as R →∞, the part of the integral associated with the circle will vanish. This is also true for λ = q + 1, provided only that α < 0, which is important for what follows.Google Scholar
  14. 1.
    Cf. E. Lindelöf [1]. In the case λ = q, α + 1 = 0, one finds by means of a special elementary estimate that log f(z; q, — 1) ~ log log z.Google Scholar
  15. 1.
    Cf. e.g. H. Lebescue [1].Google Scholar
  16. 1.
    The brothers F. and M. Riesz proved this theorem for the special case of a bounded function [1].Google Scholar
  17. 2.
    The following theorem was proved independently by O. Frostman [1] and the author [13] as an extension of an older theorem (R. Nevanlinna [7]).Google Scholar
  18. 1.
    Sharper forms of boundary value theorems have been given for special classcs of bounded functions and of functions of bounded type by O. Frostman [4] and L. Carleson [1], The boundary behavior of meromorphic functions under more general hypotheses has been investigated by E. F. Collingwood and M. L. Cartwright [1].Google Scholar
  19. 1.
    R. Nevanlinna [12].Google Scholar
  20. 1.
    Also cf. P. J. Myrberg [1].Google Scholar
  21. 1.
    R. Nevanlinna [12].Google Scholar
  22. 1.
    Taking into consideration the theorems proved above, according to which a function of bounded type has boundary values almost everywhere that form a set of positive inner capacity, the value distribution of functions of bounded type has been investigated by O. Lehto [2].Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1970

Authors and Affiliations

  • Rolf Nevanlinna
    • 1
  1. 1.Academy of FinlandFinland

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