Extremum Problems for Bounded Univalent Functions
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646 alii Tammi
Extremum Problems for Bounded Univalent Functions
Springer-Verlag Berlin Heidelberg New York 1978
Author OlliTammi Department of Mathematics University of Helsinki Hallituskatu 15 SF-DOl 00 Helsinki 10 Finland
AMS Subject Classifications (1970): 30A34, 30A38, 30A40 ISBN 3-540-08756-7 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08756-7 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 1978 Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Preface
This treatise is based on my seminar lectures and resulting discussions in Helsinki University during the academic years 1974-1977.
It is my aim
to give a survey of the main methods concerning univalent functions, i.e. the use of Lowner's functions, Schiffer's differential equations and Grunsky type inequalities.
These shall be tested by deriving certain coefficient results.
It appears especially, that some basic information concerning coefficient bodies can be found from defined Grunsky type conditions.
On the other hand, Schiffer's
differential equation characterizes all boundary functions of the coefficient body.
Hence, a natural problem in the field of inequalities arises: The Grunsky
type inequalities must be so extended that they are sharp for these Schifferfunctions.
One possible method here is the use of Lowner's functions.
This
survey is an introduction to that task. Only those works that are directly related to the theme are given in the Reference list. I wish to express my gratitude to Mr. H. Haario and O. Jokinen for their valuable contribution in preparing and reading the manuscript.
Helsinki, March 1977
Olli Tammi
Contents
I.
GENERATED FUNCTIONS
§.
I nt r oducti on
1.
The Classes
2.
On Sets and Sequences of Functions
2 §.
1.
Sand
S(b)
Lowner- functions
6
One- s l it Discs and Thei r Kerne l
8
3.
The Sol utions
Differential Equati on of One-slit Functions E S(e- t )
17
4.
Lowner ' s Equation of Second Kind
19
5.
Determining the Coe f ficients
2.
21
6 . Integrated Recursion Formulae for a n 7. The Coefficients of the Inverse Function 8. Comparison of the Coeffi cients a and ex n n 3 §.
24 27
29
Generalized Lowner-funct ions On an Extremum Func t ion of
38
2.
a 3 (b) Replac i ng the Generating Function by a Piecewice Cont i nuous Function
41
3.
The Class of Slit-functions G iven by Step-functions
42 44
1.
4.
Sequence and Extr emum Func t ion
5·
An Effort to Vary an Ar bitrary Function by Aid of Approximation
6. 7. 8.
Variation of
The Extremum
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