Subharmonic functions, vol.2

Cover......Page 1
Series......Page 2
Title page......Page 3
Copyright page......Page 4
Preface to Volume 2......Page 5
Corrections to Volume 1......Page 14
Acknowledgements......Page 15
Dedication......Page 16
Contents......Page 17
Contents of Volume I......Page 23
6.1 The Riesz-Herglotz Representation and the Milloux-Schmidt Inequality......Page 27
6.1.1 The Milloux-Schmidt inequality......Page 29
6.1.2 Maximum and minimum for functions of finite order......Page 33
6.2 The HKN Inequality and Kjellberg\'s Regularity Theorem......Page 35
6.2.1 An integral inequality for functions in a semidisk......Page 38
6.2.2 The Hellsten-Kjellberg-Norstad inequality......Page 41
6.2.3 Maximum and minimum of functions of order less than one......Page 44
6.2.4 An inequality for functions of order less than one......Page 46
6.2.5 Proof of Kjellberg\'s Theorem......Page 48
6.3 Further Regularity Theorems......Page 53
6.3.1 A lower bound for $v(-r)$......Page 54
6.3.2 Global behaviour of the extremals......Page 57
6.3.3 Regularity theorems for the Riesz mass and characteristic......Page 66
6.3.4 The function $\\theta(r)$ and local behaviour of the Riesz mass......Page 68
6.3.5 Density theorems......Page 73
6.3.6 The case of order one......Page 84
6.4 Cases when $C(\\mu)=1$; a Theorem of Beurling......Page 86
6.4.1 Proof of Beurling\'s Theorem......Page 88
6.4.2 An extension......Page 89
6.4.3 Minimum on a curve......Page 91
6.4.4 Some counterexamples......Page 93
6.5 The Wiman-Valiron Theory......Page 96
6.5.1 Koevari\'s test sequences......Page 98
6.5.2 Maximum modulus, maximum term and characteristic......Page 101
6.5.3 Local behaviour near the maximum and the minimum of harmonic functions......Page 104
6.5.4 Maximum, minimum and characteristic for s.h. functions......Page 107
6.5.5 Some examples......Page 113
6.6 Harmonic Functions in $\\mathbb{R}^m$......Page 116
6.7 The Minimum of Functions of Slow Growth......Page 118
6.7.1 Functions of order $(\\log r)^2$......Page 121
6.7.2 Functions of polynomial growth......Page 123
7.0 Introduction......Page 129
7.1 Thin Sets......Page 130
7.1.1 Wiener\'s criterion for the structure of thin sets......Page 137
7.1.2 Proof of Wiener\'s Theorem......Page 139
7.1.3 Wiener\'s criterion at finite points......Page 143
7.1.4 Functions having minimal growth in the plane......Page 144
7.2 Functions of Slow Growth in the Plane......Page 145
7.2.1 Some general results on $\\psi$-sequences......Page 148
7.2.2 Proof of Theorems 7.10 and 7.11......Page 152
7.2.3 Proof of Theorems 7.8 and 7.9; preliminary results......Page 156
7.2.4 Proof of Theorem 7.8......Page 161
7.2.5 Proof of Theorem 7.9......Page 163
7.3 Geometric Estimates for Capacity......Page 167
7.3.1 The capacity of an ellipsoid in $\\mathbb{R}^m$ where $m \\geq 3......Page 171
7.3.2 Capacity in $\\mathbb{R}^2$......Page 176
7.3.3 Projection and subadditivity......Page 179
7.3.4 Exceptional sets on rays......Page 183
7.3.5 Two counterexamples......Page 186
7.4 Some Applications to Function Theory......Page 191
7.4.1 Applications to $p$-valent functions......Page 195
7.5 Minimum of Functions in a Half-plane......Page 199
7.5.1 Completion of the proof of Hall\'s Theorem......Page 204
7.5.2 Some examples......Page 207
7.5.3 Boundary behaviour of $u(z)$......Page 211
7.5.4 A form of the Phragmen-Lindeloef Principle......Page 212
7.6 Boundary Behaviour in a Half-plane......Page 216
7.6.1 Sufficient conditions for a rarefied set......Page 217
7.6.2 Necessary conditions for a rarefied set......Page 223
7.6.3 Further results on rarefied sets......Page 227
7.6.4 Near-fine limits, equivalent subdomains and functions of bounded characteristic......Page 238
7.7 Boundary Behaviour in the Unit Disk......Page 244
7.7.1 Signed measures and characteristic for $\\delta$.s.h. functions......Page 247
7.7.2 Some applications......Page 252
7.7.3 Boundary behaviour of negative harmonic functions......Page 257
7.7.4 Boundary behaviour of Green\'s potentials......Page 260
7.7.5 Conclusion......Page 270
8.0 Introduction......Page 273
8.1.1 Wirtinger\'s inequality......Page 274
8.1.2 Statement of the convexity formula......Page 277
8.1.3 Proof of Theorem 8.1 when $u$ is smooth......Page 279
8.1.4 Extension to general s.h. functions......Page 282
8.1.5 Proof of Theorem 8.1 with $\\alpha_0(r)$ instead of $\\alpha(r)$......Page 284
8.1.6 Completion of proof of Theorem 8.1......Page 286
8.1.7 Inequalities for $I(r)$ and $B(r)$......Page 288
8.2 Growth and Image of Functions in the Unit Disk......Page 290
8.2.1 Proof of Theorem 8.5......Page 293
8.2.2 Growth of weakly univalent functions......Page 297
8.2.3 A condition on the area of the image......Page 299
8.2.4 Some examples......Page 302
8.3 Functions with $N$ Tracts......Page 303
8.3.1 Regularity theorems......Page 306
8.3.2 Functions with one asymptotic value......Page 315
8.3.3 Functions with asymptotic functions......Page 317
8.3.4 Preliminary results for Fenton\'s Theorem......Page 320
8.3.5 Completion of proof of Fenton\'s Theorem......Page 326
8.4 Growth on Asymptotic Paths......Page 328
8.4.1 Functions of order less than 1/2......Page 331
8.4.2 Some other results......Page 333
8.4.3 Tracts and asymptotic values......Page 334
8.5 Extremal Length......Page 337
8.5.1 Application to functions with $N$ tracts; Ahifors1 Spiral Theorem......Page 341
8.5.2 Proof of Ahifors\' Theorem......Page 347
8.5.3 Variation of the argument on a fixed circle......Page 350
8.6 Conformal-Mapping Techniques......Page 352
8.6.1 A Phragmen-Lindeloef Theorem for a strip......Page 353
8.6.2 A length area principle......Page 354
8.6.3 Ahlfors\'inequalities......Page 355
8.6.4 Eke\'s Regularity Theorem......Page 358
8.7 Regularity Theorems for the Tracts......Page 361
8.7.1 Proof of Theorem 8.27......Page 364
8.7.2 Consequences of Theorem 8.27......Page 366
8.7.3 Conclusions......Page 369
8.7.4 Examples......Page 371
8.8 Minimum on a Curve for Functions of Finite Lower Order......Page 373
8.8.1 Proof of Theorem 8.29......Page 378
8.8.2 The case $K=1$......Page 381
9.0 Introduction......Page 387
9.1 The Fundamental Theorem on the Star Function......Page 388
9.1.1 Two lemmas of Sjoegren......Page 392
9.1.2 Completion of the proof of Theorem 9.1......Page 394
9.2 Means and Symmetrization......Page 396
9.2.1 Some real-variable results......Page 397
9.2.2 Symmetrization and the Green\'s function......Page 400
9.2.3 Proof of the majorization theorem for Green\'s functions......Page 403
9.2.4 Symmetrization and harmonic measure......Page 407
9.3 Majorization Theorems for Univalent Functions......Page 411
9.3.1 Proof of Theorem 9.6......Page 414
9.3.2 Weakly univalent functions......Page 415
9.4 Conformal Mapping and the Hyperbolic Metric......Page 420
9.4.1 The hyperbolic metric......Page 424
9.4.2 Hyperbolic distances and Schottky\'s Theorem......Page 426
9.4.3 Some estimates for the hyperbolic metric......Page 430
9.5 Symmetrization and the Hyperbolic Metric......Page 434
9.5.1 Preliminary results......Page 435
9.5.3 Application to the hyperbolic metric......Page 440
9.5.4 A sharp form of Landau\'s Theorem......Page 444
9.5.5 Numerical estimates for $\\sigma(r)$ at 1 and $\\infty$......Page 446
9.5.6 Examples on Schottky\'s Theorem......Page 449
9.6 Polya Peaks and the Local Indicator for Functions in the Plane......Page 450
9.6.1 Polya peaks......Page 452
9.6.2 The Phragmen-Lindeloef indicator......Page 454
9.6.3 Proof of Theorem 9.20......Page 458
9.6.4 Uniform absolute continuity and Fuchs\'s Small Arcs Lemma......Page 463
9.6.5 An estimate for $h\'(\\theta)$......Page 469
9.7 Applications to Functions in the Plane: Paley\'s Conjecture......Page 472
9.7.1 Baernstein\'s Spread Theorem and the sum of the deficiencies......Page 475
9.7.2 The Edrei-Fuchs Ellipse Theorem......Page 480
9.8.1 Harmonic splines......Page 481
9.8.2 Examples illustrating Section 9.7......Page 484
9.9 Conclusion......Page 492
10.0 Introduction......Page 493
10.1 Minimal Positive Harmonic Functions......Page 494
10.1.1 Functions with $n$ tracts......Page 497
10.1.2 Warschawski\'s Theorem......Page 499
10.1.3 The asymptotic estimate......Page 502
10.2 Functions with Bounded Minimum......Page 507
10.2.1 Functions with $n$ tracts......Page 511
10.2.2 Functions bounded on spirals in the plane......Page 517
10.3.1 The Access Theorem......Page 521
10.3.2 The class $\\mathcal{A}$ of MacLane and Hornblower......Page 526
10.3.3 Proof that (iii)=>(i); preliminary results......Page 528
10.3.4 Completion of the proof of Theorem 10.10......Page 531
10.3.5 Some examples......Page 535
10.4 Growth Conditions for the Class $\\mathcal{A}$......Page 539
10.4.1 Construction of the domain......Page 540
10.4.2 Proof of Theorem 10.11......Page 541
10.4.3 Spiral functions and their growth......Page 544
10.4.4 Proof of Theorem 10.14; preliminary estimates......Page 547
10.4.5 Completion of the proof of Theorem 10.14......Page 550
10.4.6 Some further results......Page 553
10.5 The Kjellberg-Kennedy-Katifi approximation method......Page 554
10.5.1 Functions in the plane......Page 557
10.5.2 Functions in the unit disk......Page 560
10.5.3 Regular functions bounded on curves in the plane......Page 561
10.5.4 The spirals of Theorem 10.8......Page 566
10.5.5 A global upper bound......Page 570
10.5.6 Some examples......Page 575
10.5.7 The rate of tending to zero along the curves......Page 580
10.6 Approximation in the Unit Disk......Page 583
10.6.1 Regular functions tending to 0 and $\\infty$ on different spirals in the unit disk......Page 588
10.7 The Existence of Thin Components......Page 594
10.7.1 A function with two thin components......Page 596
10.7.2 A function with non-countably many thin components......Page 598
10.7.3 A theorem on thin components......Page 600
10.7.4 A function with one thick and two thin limit components......Page 601
References......Page 607
Index......Page 615