Subharmonic Functions (eBook)

Volume 2

W. K. Hayman (Autor)

P. M. Cohn, B. E. Johnson (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
512 Seiten
Elsevier Science (Verlag)
9781483296180 (ISBN)

Infos

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 The volume also includes some previously unpublished material.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.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 28.6.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• ISBN-13: 9781483296180 / 9781483296180