Univalent Functions-Selected Topics
- PDF / 10,179,059 Bytes
- 208 Pages / 612 x 792 pts (letter) Page_size
- 15 Downloads / 261 Views
Department of Mathematics, University of Maryland, College Park Adviser: L. Greenberg
478 Glenn Schober
Univalent Functions Selected Topics
Springer-Verlag Berlin· Heidelberg· New York 1975
Author Prof. Glenn Schober Department of Mathematics Indiana University Swain Hall East Bloomington, Indiana 47401
U.S.A
LibTary or Congress Catalogiag in PubllcaUon Data
Schober, Gl enn, 1938Uni va.lent func tions - -sel ected topics. (Lecture notes in mathematics ; 478) Bibliography: p . Includes index. L Univalent functions . I. Titl e . II. Series. QA3.L2B vol. 478 ( QA33l ] 510' .8s (515 ' .253] ISBN 0 -387- 07391-4 75-23099
AMS Subject Classifications (1970): 30A32, 30A36, 30A38, 30A40, 30A60 ISBN 3-540-07391-4 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-07391-4 Springer-Verlag New York' Heidelberg' Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1975 Offsetdruck: julius Beltz, Hemsbach/Bergstr.
PREFACE These notes are from lectures given by the author in 1973-74 at the University of Maryland, during its special year in complex analysis.
They are an attempt to bring together some basic ideas,
some new results, and some old results from a new point of view, in the theory of univalent functions. There are really two points of view that are used and intertwined in these notes.
The first is to utilize a linear space frame-
.work to study sets of univalent functions as they are situated in a space of analytic functions.
For example, in Chapter 7 we are
interested in compactness of families of univalent functions that lie in the intersection of two hyperplanes, and in Chapter 8 we are interested in their geometry in the sense of convexity theory. In the same spirit, we consider in Chapter 2 many of the special families of univalent functions and determine the extreme points of their closed convex hulls.
This point of view seems to
simplify and unify the study of their properties, for example, in solving linear extremal problems.
In keeping with this point of
view we give in Chapter 1 a derivation of the Herglotz representation based on Choquet's theorem.
In this case the route is less
elementary, but it serves to establish our point of view. The second point of view is to study extremal problems using variational considerations.
In the absence of a structural formula
for a class of functions, variational methods are a very powerful tool.
In Appendix C we incl.ude the boundary variation from the
fundamental work of M. Schiffer, and in Chapters 10 and 11 we apply
IV
it to solve some accessible problems and to give geome
Data Loading...
