Abstract
Since the characteristic function T(r) of a function meromorphic for \( \left| z \right| < R\underline{\underline < } \infty \) increases with r, the limit
certainly exists, and there are two cases to be considered, according as T(R) = ∞ or T(R) < ∞.
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
H. Poincaré [2], W. Blaschke [1].
R. Nevanlinna [3].
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∞).
P. Fatou [1].
Cf. e.g. H. Lebesgue [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.
This holds even for nontangential approach, as can be deduced from theorems which follow.
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.
Cf. E. Borel [1], G. Valiron [3].
G. Valiron [1].
As is customary, [a] stands for the largest integer ≦ a.
The value α = − 1 would require special treatment, and for the sake of brevity we exclude it from consideration.
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.
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) ~ zλ log log z.
Cf. e.g. H. Lebescue [1].
The brothers F. and M. Riesz proved this theorem for the special case of a bounded function [1].
The following theorem was proved independently by O. Frostman [1] and the author [13] as an extension of an older theorem (R. Nevanlinna [7]).
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].
R. Nevanlinna [12].
Also cf. P. J. Myrberg [1].
R. Nevanlinna [12].
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].
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). Functions of Bounded Type. 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_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-85590-0_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-85592-4
Online ISBN: 978-3-642-85590-0
eBook Packages: Springer Book Archive