Hayman / Cohn / Johnson Subharmonic Functions

Volume 2
1. Auflage 2014
ISBN: 978-1-4832-9618-0
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 512 Seiten, Web PDF

Building on the foundation laid in the first volume of Subharmonic Functions, which has become a classic, this second volume deals extensively with applications to functions of a complex variable. The material also has applications in differential equations and differential equations and differential geometry. It reflects the increasingly important role that subharmonic functions play in these areas of mathematics. The presentation goes back to the pioneering work of Ahlfors, Heins, and Kjellberg, leading to and including the more recent results of Baernstein, Weitsman, and many others. The volume also includes some previously unpublished material. It addresses mathematicians from graduate students to researchers in the field and will also appeal to physicists and electrical engineers who use these tools in their research work. The extensive preface and introductions to each chapter give readers an overview. A series of examples helps readers test their understatnding of the theory and the master the applications.

Autoren/Hrsg.

Hayman, W. K.

Cohn, P. M.

Johnson, B. E.

Inhaltsverzeichnis

1;Front Cover;1
2;Subharmonic Functions;4
3;Copyright Page;5
4;Table of Contents;18
5;Preface to Volume 2;6
6;CORRECTIONS TO VOLUME 1;15
7;Acknowledgements;16
8;Dedication;17
9;Contents of Volume 1;24
10;CHAPTER 6. Maximum and Minimum of Functions Subharmonic in the Plane;28
10.1;6.0. INTRODUCTION;28
10.2;6.1. THE RIESZ-HERGLOTZ REPRESENTATION AND THE MILLOUX-SCHMIDT INEQUALITY;28
10.3;6.2. THE HKN INEQUALITY AND KJELLBERG'S REGULARITY THEOREM;36
10.4;6.3. FURTHER REGULARITY THEOREMS;54
10.5;6.4. CASES WHEN C(µ) = 1; A THEOREM OF BEURLING;87
10.6;6.5. THE WIMAN-VALIRON THEORY;97
10.7;6.6. HARMONIC FUNCTIONS IN Rm;117
10.8;6.7. THE MINIMUM OF FUNCTIONS OF SLOW GROWTH;119
11;CHAPTER 7. Exceptional Sets;130
11.1;7.0. INTRODUCTION;130
11.2;7.1. THIN SETS;131
11.3;7.2. FUNCTIONS OF SLOW GROWTH IN THE PLANE;146
11.4;7.3. GEOMETRIC ESTIMATES FOR CAPACITY;168
11.5;7.4. SOME APPLICATIONS TO FUNCTION THEORY;192
11.6;7.5. MINIMUM OF FUNCTIONS IN A HALF-PLANE;200
11.7;7.6. BOUNDARY BEHAVIOUR IN A HALF-PLANE;217
11.8;7.7. BOUNDARY BEHAVIOUR IN THE UNIT DISK;245
12;CHAPTER 8. Tracts and Asymptotic Values of Plane Subharmonic Functions;274
12.1;8.0. INTRODUCTION;274
12.2;8.1. THE CARLEMAN-TSUJI-HEINS CONVEXITY FORMULA;275
12.3;8.2. GROWTH AND IMAGE OF FUNCTIONS IN THE UNIT DISK;291
12.4;8.3. FUNCTIONS WITH N TRACTS;304
12.5;8.4. GROWTH ON ASYMPTOTIC PATHS;329
12.6;8.5. EXTREMAL LENGTH;338
12.7;8.6. CONFORMAL-MAPPING TECHNIQUES;353
12.8;8.7. REGULARITY THEOREMS FOR THE TRACTS;362
12.9;8.8. MINIMUM ON A CURVE FOR FUNCTIONS OF FINITE LOWER ORDER;374
13;CHAPTER 9. Baernstein's Star Function and its Applications;388
13.1;9.0 INTRODUCTION;388
13.2;9.1. THE FUNDAMENTAL THEOREM ON THE STAR FUNCTION;389
13.3;9.2. MEANS AND SYMMETRIZATION;397
13.4;9.3. MAJORIZATION THEOREMS FOR UNIVALENT FUNCTIONS;412
13.5;9.4. CONFORMAL MAPPING AND THE HYPERBOLIC METRIC;421
13.6;9.5. SYMMETRIZATION AND THE HYPERBOLIC METRIC;435
13.7;9.6. PÓLYA PEAKS AND THE LOCAL INDICATOR FOR FUNCTIONS IN THE PLANE;451
13.8;9.7. APPLICATIONS TO FUNCTIONS IN THE PLANE: PALEY'S CONJECTURE;473
13.9;9.8. SOME EXAMPLES;482
13.10;9.9. CONCLUSION;493
14;CHAPTER 10. Examples of Subharmonic and Regular Functions, and the MacLane-Hornblower Class;494
14.1;10.0. INTRODUCTION;494
14.2;10.1. MINIMAL POSITIVE HARMONIC FUNCTIONS;495
14.3;10.2. FUNCTIONS WITH BOUNDED MINIMUM;508
14.4;10.3. ASYMPTOTIC PATHS AND THE MACLANE-HORNBLOWER THEORY;522
14.5;10.4. GROWTH CONDITIONS FOR THE CLASS;540
14.6;10.5. THE KJELLBERG-KENNEDY-KATIFI APPROXIMATION METHOD;555
14.7;10.6. APPROXIMATION IN THE UNIT DISK;584
14.8;10.7. THE EXISTENCE OF THIN COMPONENTS;595
15;References;608
16;Index;616