The Riemann Surface of a Univalent Function

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


Until now we have investigated single-valued functions w = w(z) for |z| < R ≦ ∞ without in general bothering with the image in the w-plane generated by the given correspondence zw. To be sure, in Chapter I we thoroughly investigated the Riemann surface associated with the special automorphic mapping function w(z; a 1,..., a q ) (the universal covering surface for the plane punctured at q points) ; and we have in the course of our discussion repeatedly made use of a more general concept, the universal covering surface of an arbitrary schlicht region in the w-plane. But even these surfaces should be considered as still very special, since the associated mapping functions are automorphic. Of course, automorphic functions and the regularly ramified Riemann surfaces associated with them very often provide the most interesting examples for the general value distribution theory. Nevertheless, it will be useful to become acquainted with a few general properties of surfaces generated by an arbitrary single-valued meromorphic function as the image of the disk |z| < R ≦ ∞. On such simply connected surfaces the inverse function z of w is well-defined and univalent, and our problem is thus to examine analytic functions characterized by these properties.


Riemann Surface Branch Point Base Point Meromorphic Function Univalent Function 
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  1. 1.
    The former is certainly the case when the surface is of bounded type.Google Scholar
  2. 1.
    The following proof is in principle due to G. Valiron [3].Google Scholar
  3. 1.
    W, Gross [2].Google Scholar
  4. 1.
    Cf. R. Nevanlinna [10], G. Elfving [1].Google Scholar
  5. 1.
    H. A. Schwarz [2].Google Scholar
  6. 1.
    According to the terminology of Montel [1], one says that the z n(w) constitute a normal family of functions.Google Scholar
  7. 1.
    We designate as a “strip” any half sheet that has only two ramified vertices, both of infinite order. In the line complex a strip is represented by a node that borders on only two elementary regions of infinite order.Google Scholar
  8. 1.
    For the analytic definition of all surfaces in the case p = 4, cf. R. Nevanlinna [9]. Certain higher order cases were later elucidated by H. Wagner [1].Google Scholar
  9. 1.
    R. Nevanlinna [6], G. Elfving [3].Google Scholar
  10. 3.
    For the symmetric surfaces mentioned this limit procedure can be handled computationally.Google Scholar
  11. 1.
    In this regard, cf.: R. Nevanlinna [9], F. Nevanlinna [3], E. Hille [1], [2], L. Ahlfors [6], G. Elfving [1], [3], H. Wagner [1], P. Laasonen [1].Google Scholar
  12. 1.
    The above problem has been essentially extended in an interesting article by E. Ullrich [4]. With the aid of a new class of Riemann surfaces (surfaces with finitely many periodic ends) Ullrich proves the existence of meromorphic functions that for finitely many points w 1, ..., w q have a prescribed rational-valued deficiency or ramification index with sum 2. For these questions the surfaces introduced by Ullrich [6] with the more general periodic (instead of logarithmic) ends are significant.Google Scholar
  13. 1.
    This theorem was conjectured in the year 1907 by Denjoy [1]. That the number of finite asymptotic values is finite and <5k was shown by Carleman [1] in a proof whose basic idea is also fundamental to Ahlfors‘s method. Also cf. Carleman [2].Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1970

Authors and Affiliations

  1. 1.Academy of FinlandFinland

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