# Meromorphic Functions of Finite Order

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

## Abstract

In the preceding discussion we have frequently observed that the degree of transcendency of a meromorphic function is reflected in the growth of its characteristic T(r). Considered as a class, meromorphic functions whose characteristic is of a definite order of magnitude have a more complicated structure the higher the order of growth. Thus, for instance, we have found that among other functions meromorphic for |z| < 1, meromorphic functions which have a bounded characteristic in the unit disk are distinguished by certain simple properties. They have, for example, definite boundary values almost everywhere on the bounding circle |z| = 1, which is not generally the case otherwise. The boundedness of the characteristic of a function meromorphic in the entire plane z ≠ ∞ implies that the function is constant. For rational functions T(r) = O(log r), while for a transcendental function the ratio T(r):log r is unbounded as r → ∞1.

## References

1. 1.
This will be proved in § 2 of this chapter.Google Scholar
2. 1.
A. Pringsheim [1], E. Lindelöf [1]. We prefer the designation “mean type” (type moyen) suggested by Lindelöf to the term “normal type” of Pringsheim.Google Scholar
3. 2.
4. 1.
F. Nevanlinna [1], K. Nevanlinna [4].Google Scholar
5. 1.
We can assume that w(0) ≠ 0, ∞. This can always be achieved by dividing the function w by a power z α of z.Google Scholar