Harmonic Mappings in the Plane

Peter Duren0521641217, 9780521641210, 9780511185106

Duren (mathematics, U. of Michigan) examines these univalent complex- valued harmonic functions of a complex variable, treating both the generalizations of univalent analytic functions and the connections with minimal surfaces. Duran’s topics include general properties of harmonic mappings, harmonic mappings into convex regions, harmonic self-mappings of the disk, harmonic univalent functions, external problems, mapping problems, minimal surfaces and curvature of minimal surfaces, with particular attention to the Weierstrass-Enneper representation. Readers may wish to be prepared in complex analysis

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Table of contents :
Cover……Page 1
Half-title……Page 3
Title……Page 5
Copyright……Page 6
Dedication……Page 7
Contents……Page 9
Preface……Page 13
1.1. Harmonic Mappings……Page 15
1.2. Some Basic Facts……Page 17
1.3. The Argument Principle……Page 21
1.4. The Dirichlet Problem……Page 25
1.5. Conformal Mappings……Page 27
1.6. Overview of Harmonic Mapping Theory……Page 30
2.1. Critical Points of Harmonic Functions……Page 32
2.2. Lewy’s Theorem……Page 34
2.3. Heinz’s Lemma……Page 35
2.4. Rado’s´ Theorem……Page 37
2.5. Counterexamples in Higher Dimensions……Page 39
2.6. Approximation Theorem……Page 41
3.1. The Radó–Kneser–Choquet Theorem……Page 43
3.2. Choquet’s Proof……Page 45
3.3. Boundary Behavior……Page 48
3.4. The Shear Construction……Page 50
3.5. Structure of Convex Mappings……Page 59
3.6. Covering Theorems and Coefficient Bounds……Page 62
3.7. Failure of the Radó–Kneser–Choquet Theorem in R……Page 68
4.1. Representation by Radó–Kneser–Choquet Theorem……Page 71
4.2. Mappings onto Regular Polygons……Page 73
4.3. Arbitrary Convex Polygons……Page 76
4.4. Sharp Form of Heinz’s Inequality……Page 80
4.5. Coefficient Estimates……Page 86
4.6. Schwarz’s Lemma for Harmonic Mappings……Page 89
5.1. Normalizations……Page 92
5.2. Normal Families……Page 93
5.3. The Harmonic Koebe Function……Page 96
5.4. Coefficient Conjectures……Page 100
6.1. Minimum Area……Page 103
6.2. Covering Theorems……Page 104
6.3. Estimation of |a2|……Page 109
6.4. Growth and Distortion……Page 111
6.5. Marty Relation……Page 115
6.6. Typically Real Functions……Page 117
6.7. Starlike Functions……Page 120
7.1. Generalized Riemann Mapping Theorem……Page 125
7.2. Collapsing……Page 126
7.3. Concavity of the Boundary……Page 129
7.4. Angles at Corners……Page 132
7.5. Existence Theorems……Page 140
7.6. Proof of Existence……Page 143
7.7. Uniqueness Problem……Page 147
8.1. Harmonic Mappings of Annuli……Page 150
8.2. Multiply Connected Domains……Page 152
8.3. Inverse of a Harmonic Mapping……Page 159
8.4. Decomposition of Harmonic Functions……Page 163
8.5. Integral Means……Page 165
9.1. Background in Surface Theory……Page 170
9.2. Isothermal Parameters……Page 176
9.3. Weierstrass–Enneper Representation……Page 179
9.4. Some Examples……Page 183
9.5. Historical Notes……Page 186
10.1. Gauss Curvature……Page 187
10.2. Minimal Graphs and Harmonic Mappings……Page 189
10.3. Heinz’s Lemma and Bounds on Curvature……Page 196
10.4. Sharp Bounds on Curvature……Page 200
10.5. Schwarzian Derivatives……Page 203
Appendix Extremal Length……Page 210
References……Page 215
Index……Page 225