Multivalent functions

MULTIVALENT FUNCTIONS, 2ND ED.......Page 1
Title Page......Page 3
Copyright Page......Page 4
Contents......Page 6
Preface......Page 10
Preface to the Second Edition......Page 12
1.1 Basic results......Page 14
1.2 Elementary growth and distortion theorems......Page 17
1.3 Means and coefficients......Page 22
1.4 Convex univalent functions......Page 24
1.5 Typically real functions......Page 26
1.6 Starlike univalent functions......Page 27
1.7 Asymptotic behaviour of the coefficients......Page 28
1.7.1 The case a=0......Page 29
1.7.2 The radius of greatest growth......Page 30
1.7.3 Behaviour on the major arc......Page 32
1.7.4 Behaviour on the minor arc......Page 34
1.7.5 Completion of the proof of Theorem 1.12......Page 37
2.0 Introduction......Page 41
2.1 A length-area principle......Page 42
2.2 The growth of multivalent functions......Page 45
2.3 Some averaging assumptions on p(R)......Page 50
2.4 Simultaneous growth near different boundary points......Page 53
2.5 Applications......Page 55
2.5.1 An example......Page 57
2.6 Functions of maximal growth......Page 58
2.7 Behaviour near the radius of greatest growth......Page 61
2.7.1 Construction of some level curves......Page 63
2.7.2 Basic estimates......Page 64
2.7.3 Proof of Theorem 2.8......Page 70
2.7.4 A bound for b......Page 72
2.7.5 Return to the unit disc......Page 73
3.0 Introduction......Page 79
3.1 The Hardy-Stein-Spencer identities......Page 80
3.2 Estimates of the means I(r)......Page 82
3.3 Estimates for the coefficients......Page 84
3.4 A counter-example......Page 89
3.5.1 The case of univalent functions......Page 91
3.5.2 The results of Carleson and Jones......Page 95
3.5.3 Baernstein\'s extension of Theorem 3.3......Page 96
3.5.4 Some auxiliary results......Page 97
3.5.5 Proof of Theorem 3.7......Page 102
3.6 Growth and omitted values......Page 107
3.7 k-symmetric functions and Szego\'s conjecture......Page 108
3.8 Power series with gaps......Page 111
4.0 Introduction......Page 116
4.1 Lipschitzian functions......Page 117
4.2 The formulae of Gauss and Green......Page 118
4.3 Harmonic functions and the problem of Dirichlet......Page 120
4.4 The Dirichlet integral and capacity......Page 122
4.4.1 Proof of Theorem 4.3......Page 123
4.5.1 Steiner symmetrization......Page 125
4.5.3 The symmetrized set O* is open......Page 126
4.5.5 The circularly symmetrized set of a domain D is a domain D*..........Page 128
4.6 Symmetrization of functions......Page 129
4.6.1 Symmetrization decreases the modulus of continuity......Page 130
4.7 Symmetrization of condensers......Page 132
4.7.1 Symmetrization decreases the Dirichlet integral......Page 133
4.8 Green\'s function and the inner radius......Page 135
4.8.1 The inner radius and conformal mapping......Page 137
4.8.2 The inner radius and symmetrization......Page 138
4.9 The principle of symmetrization......Page 140
4.10 Applications of Steiner symmetrization......Page 141
4.11 Applications of circular symmetrization......Page 143
4.12 Bounds for |f(z)| and |f\'(z)|......Page 146
4.13 Bloch\'s Theorem......Page 149
4.14 Some other results......Page 153
5.0 Introduction......Page 157
5.1 Functions without zeros......Page 158
5.2 Functions with a zero of order p at the origin......Page 161
5.3 Regularity Theorems: the case a = 0......Page 163
5.4 The case a >0: the minor arc......Page 165
5.5 The major arc......Page 167
5.6 Proof of Theorem 5.5......Page 168
5.7 Applications: the case la=1......Page 171
5.8 Functions with k-fold symmetry......Page 172
5.9 Some further results......Page 175
6.0 Introduction......Page 178
6.1 The basic formalism......Page 180
6.2 An application of Green\'s formula......Page 182
6.3 Estimates for the first term in (6.19)......Page 185
6.4 A 2-point estimate......Page 189
6.4.1 The case |f(z1)| < |f(z2)|......Page 191
6.5 Statement of the basic theorem......Page 193
6.6 Proof of Theorem 6.2......Page 196
6.7 Coefficient differences of k-symmetric functions......Page 198
6.8 Asymptotic behaviour......Page 199
6.9 Starlike functions......Page 201
6.10 The theorems of Dawei Shen......Page 204
7.0 Introduction......Page 210
7.1 Boundary behaviour in conformal mapping......Page 211
7.2 Transformations......Page 213
7.3 Structure of infinitesimal transformations......Page 216
7.4 The class S 1......Page 217
7.5 Continuity properties......Page 220
7.6 The differential equation......Page 222
7.7 Completion of proof of Theorem 7.1......Page 224
7.8 The third coefficient......Page 228
7.8.1 The Fekete-Szego Theorem......Page 230
7.9 Coefficients of the inverse functions......Page 235
7.10 The argument of f(z)/z......Page 237
7.11 Radii of convexity and starshapedness......Page 239
7.12 The argument of f\'(z)......Page 241
7.13 Conclusion......Page 242
8.0 Introduction......Page 243
8.1 Legendre polynomials......Page 244
8.2 Proof of Milin\'s conjecture: preliminary results......Page 249
8.2.1 Completion of proof......Page 251
8.2.2 An extension......Page 255
8.3 The Milin-Lebedev inequalities......Page 256
8.4 Proof of de Branges\' Theorem......Page 260
8.5 Some further results......Page 261
8.5.1 Examples......Page 266
Bibliography......Page 268
Index......Page 274