Meromorphic Functions of Finite Order

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


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.


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  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.
    G. Valiron [1].Google Scholar
  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

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© Springer-Verlag Berlin Heidelberg 1970

Authors and Affiliations

  1. 1.Academy of FinlandFinland

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