Table Of ContentGamma Lines
Asian Mathematics Series
A Series edited by Chung-Chun Yang
Department of Mathematics, The Hong Kong University of Science
and Technology, Hong Kong
Volume 1
Dynamics of Transcendental Functions
Xin-Hou Hua and Chung-Chun Yang
Volume 2
Approximate Methods and Numerical Analysis for Elliptic Complex Equations
Guo Chun Wen
Volume 3
Introduction to Statistical Methods in Modern Genetics
Mark CK Yang
Volume 4
Mathematical Theory in Periodic Plane Elasticity
Hai-Tao Cai and Jian-ke Lu
Volume 5
Gamma Lines: On the Geometry of Real and Complex Functions
Grigor A. Barsegian
G a m ma L i n es
On the Geometry of Real and
Complex Functions
Grigor A. Barsegian
Institute of Mathematics, National
Academy of Sciences of Armenia
Yerevan, Republic of Armenia
London and New York
First published 2002
by Taylor & Francis
11 New Fetter Lane, London EC4P 4EE
Simultaneously published in the USA and Canada
by Taylor & Francis Inc,
29 West 35th Street, New York, NY 10001
Taylor & Francis is an imprint of the Taylor & Francis Group
© 2002 Taylor & Francis
Publisher's Note
This book has been prepared from camera-ready copy provided by
the author.
Printed and bound in Great Britain by
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ISBN 0-415-26969-5
Contents
Introduction to the series
Preface
Introduction
1 Tangent variation principle: satellite principles
1.1. Modifications of length-area principle 4
1.2. Tangent variation principle 12
1.3. Estimates for collections of T-lines 21
1.4. Estimates of lengths of T-lines for
angular-quasiconformal mappings 29
1.5. Remarks on application of estimates ofh(D,F) 35
2 Nevanlinna and Ahlfors' theories: additions
2.1. Basic concepts and, outcomes of Nevanlinna
value distribution theory and Ahlfors' theory
of covering surfaces 40
2.2. Geometric deficient values 52
2.3. On some additions to Ahlfors' theory
of covering surfaces 64
2.4. Bounds of some integrals 75
3 T-lines' approach in the theory of meromorphic functions
3.1. Principle of closeness of sufficiently large sets
ofa-points of meromorphic functions 82
3.2. Integrated version of the principle: connections
with known classes of functions 91
vi Contents
4 Distribution of T-lines for functions meromorphic in
C: Applications 102
4.1. The main results on distribution of T-lines 102
4.2. "Windings" of T-lines 110
4.3. Average lengths of T-lines along concentric
circles and the deficient values 114
4.4. Distribution of T-lines and value distribution of modules
and real parts of meromorphic functions 117
4.5. The number of T-lines crossing rings 119
4.6. Distribution of Gelfond points 120
4.7. Nevanlinna's dream-description of transcendental
ramification ofRiemann surfaces 124
4.8. The proximity property of a-points of meromorphic
functions 134
4.9. A proof of the proximity property of a-points based
on investigation of T-lines only 140
5 Some applied problems 144
5.7. T-lines in physics 144
5.2. On the cross road of value distribution, T-lines,
free boundary theories and applied mathematics 146
5.3. "Point maps " of physical processes and a-points of
general classes of functions 156
5.4 On a-points of some non-holomorphic function 163
References 165
Index 175
Introduction to the series
The Asian Mathematics Series provides a forum to promote and reflect timely
mathematical research and development from the Asian region, and to provide
suitable and pertinent reference or text books for researchers, academics and
graduate students in Asian universities and research institutes, as well as in the
West. With the growing strength of Asian economic, scientific and technological
development, there is a need more than ever before for teaching and research
materials written by leading Asian researchers, or those who have worked in or
visited the Asian region, particularly tailored to meet the growing demands of
students and researchers in that region. Many leading mathematicians in Asia
were themselves trained in the West, and their experience with Western methods
will make these books suitable not only for an Asian audience but also for the
international mathematics community.
The Asian Mathematics Series is founded with the aim to present significant
contributions from mathematicians, written with an Asian audience in mind, to
the mathematics community. The series will cover all mathematical fields and
their applications, with volumes contributed to by international experts who have
taught or performed research in Asia. The material will be at graduate level or
above. The book series will consist mainly of monographs and lecture notes but
conference proceedings of meetings and workshops held in the Asian region will
also be considered.
Preface
The history of mathematics is in a considerable extent connected with the study
of solutions of the equation f(x) = a = const for functions f(x) for one real or
complex variable. However we knew surprizingly little about solutions of
u(x, y) = t = const for functions of two real variables. These solutions, called level
sets, are very important because of their applications in physics, environmental
problems, biology, economics, etc., as they mean a "map" of an appropriate
process described by the function u{x, y) for given parameters (x, y).
This book studies the concept of T-lines that generalizes both the classical
concepts of level sets and a-points. The author aims to show how large is the field
of possible applications of T-lines and to present a book accessible for specialists
in different sciences; at least to read Chapters 1, 3, 5 one need to be familiar only
with beginning of the standard course of complex analyses.
An interesting circumstance is that by reading these formally quite simple chapters
one can particularly get idea about leading ideas and main conclusions of classical
Value distribution theory of R. Nevanlinna that study mainly numbers of a-points
of meromorphic functions in the complex plane. Moreover, constructed theory of
Distribution of T-lines permit to transfer Value distribution theory to a new stage
of problems in studying the geometry of a-points instead of the numbers of
a-points, considered in the classical theory.
Meromorphic functions in the unit disk is less investigated. For functions of
slow growth the Second fundamental theorem of Value distribution theory do not
give efficient consequences. This is an essential gap in complex function theory
since many known classes of functions (particularly classes of Nevanlinna,
Dirichlet, Hardy) do have slow growth. The T-lines approach offered in this
book permits to describe the numbers as well as the locations of a-points of all
functions meromorphic in the unit disk.
On the other hand, we study solutions of the following system
u(x,y) = t=const, \grad u(x,y)\=R = const, for some classes of functions u(x,y)
associated with meromorphic functions. The existence of the solutions for different
classes of u(x, y) is studied in the theory of Free Boundary Problems which
Preface ix
thanks to rich applied content is one of the recent hot topics. Using T-lines we are
able to describe the numbers of these solutions for a class of functions u(x, y). To our
surprise we find a kind of Nevanlinna Deficiency Relation for this distinct topic.
The T-lines of some particular functions classes as well as applications of level
sets in physics and engineering were subjects of very active international
research in the last 15 years. The author hopes that the methods of the present
book will be useful for the development of the mentioned and of new topics
related both to pure mathematics and applications. At first this concern a new and
wide program: the study of level sets for solutions of partial differential equations
(and other type equations). Physical interpretations of these level sets and
respectively expected bearings in applications, seem to be evident.
My pleasant duty is thanking people for their attention and valuable discussions
while attaining the book results: Professors C. Andreian-Cazacu, N. Arakelian,
A. Beliy, W.H.J. Fuchs, A. Goldberg, A. Gontchar, S. Mergelian, G. Suvorov.
I am greatly indebted to Professors H. Begehr and I. Laine for carefully
reading the book manuscript and suggesting several improvements.
My special thanks go to Professor C.C. Yang and Dr G. Sukiasian for their
patience and invaluable help during the final stage of preparation of this book.
YEREVAN, 1999
