Extremum Problems for Bounded Univalent Functions II
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913 alii Tammi
Extremum Problems for Bounded Univalent Functions II
Springer-Verlag Berlin Heidelberg New York 1982
Author
OlliTammi Department of Mathematics, University of Helsinki Hallituskatu 15, 00100 Helsinki 10, Finland
AMS Subject Classifications (1980): 30C20, 30C50, 30C55, 30C75. ISBN 3-540-11200-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11200-6 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 "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Preface "Extremum Problems for Bounded Univalent Functions", Volume 646 of Lecture Notes in Mathematics, was published in 1978.
Its aim was to found such generalized
Grunsky type inequalities which allow direct sharp estimation of functionals so chosen that equality is reached by certain prescribed solutions of Schiffer's differential equation. mentioned.
Volume 646 constitutes an introduction to the problematics
Actually, there are no tests in it which could indicate the effective-
ness of the ideas proposed. The present work is devoted to the tests and is thus a continuation to the previous one.
The results are due to discussions with colleagues and students
belonging to our research group in Helsinki.
The present development has
benefited ln an essential way from the ideas of H. Haario, O. Jokinen and
R. Kortram, as can be seen from the short reference list. The estimation technique developed appears to be effective In problems (a of bounded 2,a 3) seems to be just on The next body, (a , 2,a 3,a4) the limit of the range of effectiveness of our tool. In the real subclass connected with the first nontrivial coefficient body
un i va.Lerrt
SR(b)
of
functions S(b)
S(b) .
one finds a complete characterization of the algebraic part
of the coefficient body. determined in
SR(b)
As a consequence of this the maximum of
for all values of
a4
can be
b.
The present computer technique allows illustrating some of the results by graphs unattainable by purely manual computations.
I am grateful for these
graphs to O. Jokinen who skillfully composed the drawing programs involved.
Helsinki, March 1981 ani Tammi
Content s V. 1
THE FIRST COEFFICIENT BODY AND RELATED PROBLEMS IN
§.
Determination of the Coefficient Body
(a
a Perfect Square Method
2
3
4
2,a3)
S(O)
by Aid of
1.
Introduction
2.
The Boundary Functions
2:2
2
3.
The Boundary Functions
1:2
12
4.
The Boundary Functions
1:1
26
§.
The Linear Combination
a
1.
min (a
2.
max (a
3
+ Aa
2)
+ Aa
3.
2) max (a + Aa 2) 3
§.
Re (a
3
for for
e
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