Analytic Functions pp 181-213 | Cite as

# Functions of Bounded Type

Chapter

## Abstract

Since the characteristic function certainly exists, and there are two cases to be considered, according as

*T*(*r*) of a function meromorphic for \( \left| z \right| < R\underline{\underline < } \infty \) increases with*r*, the limit$$
T\left( R \right) = \mathop {\lim }\limits_{r = R} T\left( r \right)
$$

*T*(*R*) = ∞ or*T*(*R*) < ∞.### Keywords

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### References

- 1.
- 2.R. Nevanlinna [3].Google Scholar
- 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 - 1.P. Fatou [1].Google Scholar
- 2.Cf. e.g. H. Lebesgue [1].Google Scholar
- 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 - 1.This holds even for nontangential approach, as can be deduced from theorems which follow.Google Scholar
- 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 - 1.
- 1.G. Valiron [1].Google Scholar
- 1.As is customary, [a] stands for the largest integer ≦
*a*.Google Scholar - 1.The value
*α*= − 1 would require special treatment, and for the sake of brevity we exclude it from consideration.Google Scholar - 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 - 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) ~*zλ*log log*z*.Google Scholar - 1.Cf. e.g. H. Lebescue [1].Google Scholar
- 1.The brothers F. and M. Riesz proved this theorem for the special case of a bounded function [1].Google Scholar
- 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
- 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
- 1.R. Nevanlinna [12].Google Scholar
- 1.Also cf. P. J. Myrberg [1].Google Scholar
- 1.R. Nevanlinna [12].Google Scholar
- 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

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