Geometric Function Theory in One and Higher Dimensions

Geometric Function Theory in One and Higher Dimensions......Page 1
Preface......Page 10
Introduction......Page 18
Contents......Page 12
1.1.1 Elementary results in the theory of univalent functions. Examples of univalent functions......Page 23
1.1.2 The area theorem......Page 29
1.1.3 Growth, covering and distortion results in the class S......Page 33
1.1.4 The maximum modulus of univalent functions......Page 38
1.1.5 Two-point distortion results for the class S......Page 41
Problems......Page 44
2.1.1 The Caratheodory class. Subordination......Page 47
2.1.2 Applications of the subordination principle......Page 52
Problems......Page 54
2.2 Starlike and convex functions......Page 56
Problems......Page 72
2.3.1 Starlikeness and convexity of order alpha......Page 74
2.3.2 Alpha convexity......Page 78
Problems......Page 81
2.4.1 Close-to-convexity in the unit disc......Page 83
2.4.2 Spirallike functions in the unit disc......Page 93
2.4.3 Phi-like functions on the unit disc......Page 99
Problems......Page 103
3.1.1 Kernel convergence......Page 106
3.1.2 Subordination chains and kernel convergence......Page 113
3.1.3 Loewner\'s differential equation......Page 119
3.1.4 Remarks on Bieberbach\'s conjecture......Page 131
Problems......Page 135
3.2 Applications of Loewner\'s differential equation to the study of univalent functions......Page 136
3.2.1 The radius of starlikeness for the class S and the rotation theorem......Page 137
3.2.2 Applications of the method of Loewner chains to characterize some subclasses of S......Page 145
3.3.1 Becker\'s univalence criteria......Page 149
3.3.2 Univalence criteria involving the Schwarzian derivative......Page 151
3.3.3 A generalization of Becker\'s and Nehari\'s univalence criteria......Page 159
Problems......Page 162
4.1 Preliminaries concerning Bloch functions......Page 163
4.2 The Bloch constant problem and Bonk\'s distortion theorem......Page 169
4.3.1 Distortion results for locally univalent Bloch functions......Page 175
4.3.2 The case of convex functions......Page 181
Problems......Page 182
5.1 General ideas concerning linear-invariant families......Page 183
5.2.1 Bounds for coefficients of functions in linear-invariant families......Page 190
5.2.2 Radius problems for linear-invariant families......Page 192
Problems......Page 195
Chapter 6: Univalence in several complex variables......Page 199
6.1.1 Holomorphic functions in Cn......Page 200
6.1.2 Classes of domains in Cn. Pseudoconvexity......Page 204
6.1.3 Holomorphic mappings......Page 207
6.1.4 Automorphisms of the Euclidean unit ball and the unit polydisc......Page 211
6.1.5 Holomorphic mappings in complex Banach spaces......Page 213
6.1.6 Generalizations of functions with positive real part......Page 218
6.1.7 Examples and counterexamples......Page 226
6.2.1 Criteria for starlikeness on the unit ball in Cn or in a complex Banach space......Page 229
6.2.2 Starlikeness criteria on more general domains in Cn......Page 233
6.2.3 Sufficient conditions for starlikeness for mappings of class C1......Page 235
6.2.4 Starlikeness of order gamma in Cn......Page 237
Problems......Page 238
6.3.1 Criteria for convexity on the unit polydisc and the Euclidean unit ball......Page 239
6.3.2 Necessary and sufficient conditions for convexity in complex Banach spaces......Page 246
6.3.3 Quasi-convex mappings on the unit ball of Cn......Page 254
Problems......Page 258
6.4 Spirallikeness and phi-likeness in several complex variables......Page 260
Problems......Page 268
Chapter 7: Growth, covering and distortion results for starlike and convex mappings in Cn and complex Banach spaces......Page 270
7.1.1 Growth and covering results for starlike mappings on the unit ball and some pseudoconvex domains in Cn. Extensions to complex Banach spaces......Page 271
7.1.2 Bounds for coefficients of normalized starlike mappings in Cn......Page 277
Problems......Page 282
7.1.3 A distortion result for a subclass of starlike mappings in Cn......Page 283
7.2.1 Growth and covering results for convex mappings......Page 286
7.2.2 Covering theorem and the translation theorem in the case of nonunivalent convex mappings in several complex variables......Page 293
7.2.3 Bounds for coefficients of convex mappings in Cn and complex Hilbert spaces......Page 296
7.2.4 Distortion results for convex mappings in Cn and complex Hilbert spaces......Page 301
Problems......Page 309
8.1.1 The Loewner differential equation in Cn......Page 310
8.1.2 Transition mappings associated to Loewner chains on the unit ball of Cn......Page 327
Problems......Page 335
8.2.1 An alternative characterization of spirallikeness of type alpha in terms of Loewner chains......Page 337
8.2.2 Close-to-starlike mappings on the unit ball of Cn......Page 339
8.3.1 Examples of mappings which admit parametric representation on the unit ball of Cn......Page 345
8.3.2 Growth results and coefficient bounds for mappings in S 0 g(B)......Page 349
8.4 Applications of the method of Loewner chains to univalence criteria on the unit ball of Cn......Page 363
Problems......Page 367
8.5.1 Construction of quasiconformal extensions by means of Loewner chains......Page 368
8.5.2 Strongly starlike and strongly spirallike mappings of type a on the unit ball of Cn......Page 385
Problems......Page 388
9.1 Preliminaries and a generalization of Bonk\'s distortion theorem......Page 391
9.2 Bloch constants for bounded and quasiregular holomorphic mappings......Page 398
9.3 Bloch constants for starlike and convex mappings in several complex variables......Page 404
Chapter 10: Linear invariance in several complex variables......Page 408
10.1.1 L.I.F.\'s and trace order in several complex variables......Page 409
10.1.2 Examples of L.l.F.\'s on the Euclidean unit ball of Cn......Page 412
10.2.1 Distortion results for L.I.F. \'s on the Euclidean unit ball of Cn......Page 414
10.2.2 Distortion results for L.I.F.\'s on the unit polydisc of Cn......Page 423
Problems......Page 426
10.3.1 Examples of L.I.F.\'s of minimum order on the Euclidean unit ball of Cn......Page 427
10.3.2 Examples of L.I.F. \'s of minimum order on the unit polydisc of Cn......Page 439
Problems......Page 441
10.4 Norm order of linear-invariant families in several complex variables......Page 442
Problems......Page 446
10.5 Norm order and univalence on the Euclidean unit ball of Cn......Page 447
Problems......Page 452
10.6 Linear-invariant families in complex Hilbert spaces......Page 453
Problems......Page 454
Chapter 11: Univalent mappings and the Roper-Suffridge extension operator......Page 455
11.1 Convex, starlike and Bloch mappings and the Roper-Suffridge extension operator......Page 456
11.2 Growth and covering theorems associated with the Roper-Suffridge extension operator......Page 468
11.3 Loewner chains and the operator phi n,alpha......Page 473
11.4 Radius problems and the operator phi n,alpha......Page 478
11.5 Linear-invariant families and the operator phi n,alpha......Page 481
Problems......Page 486
Bibliography......Page 488
Symbols in Chapter 2......Page 532
Symbols in Chapter 4......Page 533
Symbols in Chapter 6......Page 534
Symbols in Chapter 7......Page 535
Symbols in Chapter 9......Page 536
Symbols in Chapter 11......Page 537