Handbook of Complex Analysis

Description

Geometric Function Theory is a central part of Complex Analysis (one complex variable). The Handbook of Complex Analysis - Geometric Function Theory deals with this field and its many ramifications and relations to other areas of mathematics and physics. The theory of conformal and quasiconformal mappings plays a central role in this Handbook, for example a priori-estimates for these mappings which arise from solving extremal problems, and constructive methods are considered. As a new field the theory of circle packings which goes back to P. Koebe is included. The Handbook should be useful for experts as well as for mathematicians working in other areas, as well as for physicists and engineers.

· A collection of independent survey articles in the field of GeometricFunction Theory
· Existence theorems and qualitative properties of conformal and quasiconformal mappings
· A bibliography, including many hints to applications in electrostatics, heat conduction, potential flows (in the plane)

Cover 2
Handbook of Complex Analysis: Geometric Function Theory 5
Copyright 6
Preface 7
List of Contributors 11
Contents 13
Chapter 1. Univalent and multivalent functions 15
- 1. Univalent functions 17
- 2. Asymptotic behaviour 25
- 3. Löwner's theory and de Branges' theorem 31
- 4. Subclasses 37
- 5. Brennan's conjecture and related problems 39
- 6. Multivalent functions 40
- References 48
Chapter 2. Conformal maps at the boundary 51

Cover 2
Handbook of Complex Analysis: Geometric Function Theory 5
Copyright 6
Preface 7
List of Contributors 11
Contents 13
Chapter 1. Univalent and multivalent functions 15
- 1. Univalent functions 17
- 2. Asymptotic behaviour 25
- 3. Löwner's theory and de Branges' theorem 31
- 4. Subclasses 37
- 5. Brennan's conjecture and related problems 39
- 6. Multivalent functions 40
- References 48
Chapter 2. Conformal maps at the boundary 51
- 1. Introduction 53
- 2. Continuity at the boundary 55
- 3. Domains with nice boundaries 63
- 4. General boundaries 69
- 5. Some special types of domains 78
- References 84
- Acknowledgements 84
Chapter 3. Extremal quasiconformal mappings of the disk 89
- 1. Introduction 91
- Preface 91
- 2. The Hamilton–Krushkal condition – necessity 103
- 3. The main inequality 111
- 4. Local versus global effects 118
- 5. Unique extremality 125
- 6. The case of infinitesimal dilatations 140
- Acknowledgements 143
- References 144
Chapter 4. Conformal welding 151
- 1. Introduction 153
- 2. Existence 154
- 3. Uniqueness 157
- 4. Fuchsian groups 158
- 5. Regularity 159
- References 159
Chapter 5. Area distortion of quasiconformal mappings 161
- 1. Introduction 163
- 2. From Grötzsch to Bojarski 164
- 3. Holomorphy 166
- 4. The class Σ(k) 167
- 5. A strange Harnack inequality 168
- 6. Theorem 1, part I 169
- 7. Theorem 1, part 2 170
- 8. Bounds on the Beurling–Ahlfors transform 172
- 9. Further applications 173
- References 173
Chapter 6. Siegel disks and geometric function theory in the work of Yoccoz 175
- 1. Introduction 177
- 2. The theorem of Yoccoz 178
- References 181
Chapter 7. Sufficient conditions for univalence and quasiconformal extendibility of analytic functions 183
- 1. Introduction and classification of univalence conditions 185
- 2. A review of sufficient conditions for quasiconformal extendibility in canonical domains 188
- 3. Conditions for quasiconformal extension in non-canonical domains 204
- 4. Univalence conditions in multiply connected domains 208
- 5. Certain relations with universal Teichmüller spaces 211
- 6. Mechanical and physical applications 213
- References 217
Chapter 8. Bounded univalent functions 221
- 1. Introduction 223
- 2. Coefficient growth in S(M) 224
- 3. Parametric representations of bounded univalent functions 224
- 4. Control theory methods in extremal problems for S(M) 227
- 5. Coefficient estimates 233
- 6. Subclasses of bounded univalent functions 234
- References 240
Chapter 9. The *-function in complex analysis 243
- 1. Introduction 245
- 2. General properties of the *-function 247
- 3. Subharmonicity properties of the *-function 248
- 4. Nevanlinna's N function 253
- 5. Integral means of univalent functions 255
- 6. Circular symmetrization, Green functions, and harmonic measures 258
- 7. Vertical *-functions and Steiner symmetrization 261
- 8. Conjugate harmonic functions 264
- 9. Variants of the *-function 268
- 10. The spread relation 271
- 11. Paley's conjecture 276
- 12. Symmetrization and the hyperbolic metric 278
- References 282
Chapter 10. Logarithmic geometry, exponentiation, and coefficient bounds in the theory of univalent functions and nonoverlappin 287
- 1. Introduction 289
- 2. Full mappings and slit mappings 290
- 3. The area method 291
- 4. The Bieberbach theorem and conjecture 293
- 5. The Löwner method: parametric representation of slit mappings 294
- 6. The Grunsky univalence criterion and Milin area theorem 296
- 7. The Lebedev area theorem for nonoverlapping domains 301
- 8. Quasiconformal and regularly measurable maps 308
- 9. The Grunsky operator and quasiconformally extendible functions 310
- 10. Pairs of functions with nonoverlapping image domains and the τ-norm 313
- 11. Exponentiation, Milin's method, and descent technique 316
- 12. The Milin theorem and conjecture on logarithmic functionals 324
- 13. De Branges' theorem and the proof of the Bieberbach conjecture 325
- 14. Successive coefficients of univalent functions and coefficients of odd functions 328
- 15. Estimates for functions whose range has a finite logarithmic area 330
- 16. Coefficient properties of Gel' fer functions 332
- 17. Coefficient estimates depending on Grunsky norm 333
- 18. The Goodman conjecture and polynomial compositions 335
- 19. A remark on noncoefficient problems 336
- References 337
Chapter 11. Circle packing and discrete analytic function theory 347
- Abstract 347
- Introduction 349
- 1. Discrete analytic function theory 351
- 2. Discrete conformal geometry 371
- Appendix A: Computational notes 379
- Appendix B: The literature 380
- References 381
Chapter 12. Extreme points and support points 385
- 1. Introduction 387
- 2. Integral representations 388
- 3. Subclasses of S 389
- 4. Subordination classes 392
- 5. Applications 396
- 6. Univalent functions 400
- 7. Notes 403
- References 404
Chapter 13. The method of the extremal metric 407
- Abstract 407
- 1. History, definitions and standard results 409
- 2. Quadratic differentials 414
- 3. Modules of multiple curve families 419
- 4. The General Coefficient Theorem 424
- 5. Symmetrization 428
- 6. Boundary correspondence. Boundary distortion 431
- 7. Univalent regular and meromorphic functions and other function families 433
- 8. Strip domains. Angular derivatives 440
- 9. Harmonic measures and triad modules 445
- 10. Application to non-univalent functions 448
- 11. Multivalent functions 453
- 12. Quasiconformal mappings 456
- 13. Various results 460
- Acknowledgments 463
- References 463
Chapter 14. Universal Teichmüller space 471
- Abstract 472
- Introduction 473
- 1. Real analysis 476
- 2. Complex analysis 489
- References 504
Chapter 15. Application of conformal and quasiconformal mappings and their properties in approximation theory 507
- Abstract 507
- 1. Polynomial approximation on general continua 509
- 2. Classes of sets 511
- 3. Uniform approximation. Approximation characteristic 515
- 4. Dzjadyk type approximation 518
- 6. Approximation by harmonic polynomials 527
- References 533
Author Index 535
Subject Index 545

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