Geometric theory of functions of a complex variable (Translations of Mathematical Monographs, Vol. 26)

Cover......Page 1
Geometric Theory of Functions of a Complex Variable......Page 2
ISBN 821815760......Page 3
Contents......Page 4
A note on the author......Page 8
Preface to the second edition......Page 10
Preface (to the first edition)......Page 11
Introductory geometric considerations                        ......Page 12
1. The convergence of sequences of analytic functions......Page 18
2. The condensation principle......Page 21
1. Univalent conformal mapping......Page 30
2. Riemann\'s theorem......Page 32
3. The correspondence of boundaries under conformal mapping......Page 38
4. Distortion theorems......Page 54
5. Convergence theorems on the conformal mapping of a sequence of domains......Page 61
6. Modular and automorphic functions......Page 69
7. Normal families of analytic functions. Applications......Page 74
1. Conformal mapping of domains bounded by rectilinear and circular polygons......Page 83
2. Parametric representation of univalent functions......Page 96
3. Variation of univalent functions......Page 106
1. Rotation theorems......Page 117
2. Sharpening of the distortion theorems......Page 125
3. Extrema and majorizations of the type of the distortion theorems......Page 135
4. Application of the method of variations to other extremal problems......Page 147
5. Limits of convexity and starlikeness......Page 172
6. Covering of segments and areas......Page 177
7. Lemmas on the mean modulus. Bounds for the coefficients......Page 189
8. The relative growth of coefficients of univalent functions......Page 197
9. Sharp bounds on the coefficients......Page 203
1. Univalent conformal mapping of a doubly connected domain onto an annulus......Page 212
2. Univalent mapping of a multiply connected domain onto a plane with parallel rectilinear cuts......Page 217
3. Univalent mapping of a multiply connected domain onto a helical domain......Page 223
4. Some relationships involving the mapping functions......Page 229
5. Convergence theorems for univalent mapping of a sequence of domains......Page 235
6. Univalent mapping of multiply connected domains onto circular domains. The continuity method......Page 241
7. Proof of Brouwer\'s theorem......Page 251
1. Conformal mapping of a multiply connected domain onto a disk......Page 261
2. Correspondence of boundaries under a mapping of a multiply connected domain onto a disk......Page 269
3. Dirichlet\'s problem and Green\'s function......Page 273
4. Application to a univalent mapping of multiply connected domains......Page 282
5. Mapping of an n-connected domain onto an n-sheeted disk......Page 284
6. Some identities connecting a univalent conformal mapping and the Dirichlet problem......Page 290
1. The transfinite diameter and Cebysvev\'s constant......Page 300
2. Bounds for the transfinite diameter......Page 307
3. The capacity of a closed bounded set......Page 316
4. Harmonic measure of closed bounded sets......Page 321
5. An application to meromorphic functions of bounded form......Page 328
1. An invariant form of the Schwarz lemma......Page 336
2. The hyperbolic metric principle......Page 343
3. Lindeldf\'s principle......Page 346
4. Harmonic measure. The simplest applications......Page 348
5. On the number of asymptotic values of entire functions of finite order......Page 358
6. The hyperconvergence of power series......Page 363
7. A nonanalytic generalization of the Schwarz lemma. A theorem on covering of disks......Page 367
8. Majorization of subordinate analytic functions......Page 375
1. Limiting values of Poisson\'s integral......Page 387
2. The representation of harmonic functions by means of Poisson\'s integral and the Poisson-Stieltjes integral......Page 392
3. The limiting values of analytic functions......Page 400
4. Boundary properties of functions in the class Hp......Page 409
5. Functions that are continuous on a closed disk......Page 416
1. The correspondence of boundaries under conformal mapping......Page 424
2. Privalov\'s uniqueness theorem......Page 435
3. On the limiting values of Cauchy\'s integral......Page 437
4. Cauchy\'s formula......Page 442
5. Classes of functions. Cauchy\'s formula......Page 445
6. On the extrema of mean moduli......Page 448
7. Approximation in mean and the theory of orthogonal polynomials......Page 455
1. Gluing theorems......Page 461
2. Conformal mapping of simply connected Riemann surfaces......Page 468
3. An extremum for bounded functions in multiply connected domains......Page 474
4. The three-disk theorem......Page 483
5. Transformation of analytic functions by means of polynomials......Page 487
6. On p-valent functions......Page 494
7. Some remarks on the Caratheodory-Fejer problem and on an analogous problem......Page 504
8. Some inequalities for bounded functions......Page 521
9. A method of variations in the. theory of analytic functions......Page 533
10. The scientific works of Gennadii Mihailovic Goluzin......Page 552
Bibliography......Page 556
Introduction......Page 570
1. Basic methods of the geometric theory of functions of a complex variable......Page 572
2. Univalent functions in a disk and in an annulus......Page 584
3. Functions that are analytic in multiply connected domains......Page 636
Bibliography for the Supplement......Page 658
Subject Index......Page 680
Back Cover......Page 684