Analytic Functions pp 282-307 | Cite as

# The Riemann Surface of a Univalent Function

## Abstract

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 *z* → *w*. 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.

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- 1.The former is certainly the case when the surface is of bounded type.Google Scholar
- 1.The following proof is in principle due to G. Valiron [3].Google Scholar
- 1.W, Gross [2].Google Scholar
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- 1.H. A. Schwarz [2].Google Scholar
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*z*_{n}(*w*) constitute a*normal*family of functions.Google Scholar - 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
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*p*= 4, cf. R. Nevanlinna [9]. Certain higher order cases were later elucidated by H. Wagner [1].Google Scholar - 1.
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- 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 - 1.This theorem was conjectured in the year 1907 by Denjoy [1]. That the number of finite asymptotic values is finite and <5
*k*was shown by Carleman [1] in a proof whose basic idea is also fundamental to Ahlfors‘s method. Also cf. Carleman [2].Google Scholar