Vol. 25, No. 1, 1968

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 329: 1
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Online Archive
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author index
To appear
Other MSP journals
A note on extremal properties characterizing weakly λ-valent principal functions

Dennis Garoutte and Paul Adrian Nickel

Vol. 25 (1968), No. 1, 109–115

On a planar bordered Riemann surface W, a weakly λ-valent function is one whose every image point has at most λ antiimages. In this note, extremal properties characterizing weakly λ-valent principal functions are developed. The functionals extremized are, in a rather natural way, analogous to those of the univalent cases. However, the class of competing functions consists not only of weakly λ-valent analytic functions on W, but of all analytic functions which are λ-valent near an interior point ζ W and near the isolated border γ of W, and are of arbitrary finite valence elsewhere. Such competing classes contain the λ-th powers of competing univalent functions, as would be expected. That these classes contain functions of arbitrary finite valence perhaps would not be anticipated.

An interpretation is given for that situation in which the competing classes consist of those analytic functions which are λ-valent near two isolated border components.

Mathematical Subject Classification
Primary: 30.44
Received: 6 February 1967
Published: 1 April 1968
Dennis Garoutte
Paul Adrian Nickel