## Geometric Theory of Functions of a Complex Variable |

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### Contents

5 | |

11 | |

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19 | |

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25 | |

31 | |

4 Distortion theorems | 47 |

The measure properties of closed sets in the plane | 293 |

2 Bounds for the transfinite diameter | 300 |

3 The capacity of a closed bounded set | 309 |

4 Harmonic measure of closed bounded sets | 314 |

5 An application to meromorphic functions of bounded form | 321 |

Majorization principles and their applications | 329 |

2 The hyperbolic metric principle | 336 |

3 Lindelofs principle | 339 |

5 Convergence theorems on the conformal mapping of a sequence of domains | 54 |

6 Modular and automorphic functions | 62 |

7 Normal families of analytic functions Applications | 67 |

Realization of conformal mapping of simply connected domains | 76 |

2 Parametric representation of univalent functions | 89 |

3 Variation of univalent functions | 99 |

Extremal questions and inequalities holding in classes of univalent functions | 110 |

2 Sharpening of the distortion theorems | 118 |

3 Extrema and majorizations of the type of the distortion theorems | 128 |

4 Application of the method of variations to other extremal problems | 140 |

5 Limits of convexity and starlikeness | 165 |

6 Covering of segments and areas | 170 |

7 Lemmas on the mean modulus Bounds for the coefficients | 182 |

8 The relative growth of coefficients of univalent functions | 190 |

9 Sharp bounds on the coefficients | 196 |

Univalent conformal mapping of multiply connected domains | 205 |

2 Univalent mapping of a multiply connected domain onto a plane with parallel rectilinear cuts | 210 |

3 Univalent mapping of a multiply connected domain onto a helical domain | 216 |

4 Some relationships involving the mapping functions | 222 |

5 Convergence theorems for univalent mapping of a sequence of domains | 228 |

6 Univalent mapping of multiply connected domains onto circular domains The continuity method | 234 |

7 Proof of Bronwers theorem | 244 |

Mapping of multiply connected domains onto a disk | 254 |

2 Correspondence of boundaries under a mapping of a multiply connected domain onto a disk | 262 |

3 Dirichlets problem and Greens function | 266 |

4 Application to a univalent mapping of multiply connected domains | 275 |

5 Mapping of an nconnected domain onto an nsheeted disk | 277 |

6 Some identities connecting a univalent conformal mapping and the Dirichlet problem | 283 |

4 Harmonic measure The simplest applications | 341 |

5 On the number of asymptotic values of entire functions of finite order | 351 |

6 The hyperconvergence of power series | 356 |

7 A nonanalytic generalization of the Schwarz lemma A theorem on covering of disks | 360 |

8 Majorization of subordinate analytic functions | 368 |

Boundary value problems for analytic functions defined on a disk | 380 |

2 The representation of harmonic functions by means of Poissons integral and the PoissonStieltjes integral | 385 |

3 The limiting values of analytic functions | 393 |

4 Boundary properties of functions in the class Hₚ | 402 |

5 Functions that are continuous on a closed disk | 409 |

Boundary questions for functions that are analytic inside a rectifiable contour | 417 |

2 Privalovs uniqueness theorem | 428 |

3 On the limiting values of Cauchys integral | 430 |

4 Cauchys formula | 435 |

5 Classes of functions Cauchys formula | 438 |

6 On the extrema of mean moduli | 441 |

7 Approximation in mean and the theory of orthogonal polynomials | 448 |

Some supplementary information | 454 |

2 Conformal mapping of simply connected Riemann surfaces | 461 |

3 An extremum for bounded functions in multiply connected domains | 467 |

4 The threedisk theorem | 476 |

5 Transformation of analytic functions by means of polynomials | 480 |

6 On pvalent functions | 487 |

7 Some remarks on the CaratheodoryFejér problem and on an analogous problem | 497 |

8 Some inequalities for bounded functions | 514 |

9 A method of variations in the theory of analytic functions | 526 |

### Other editions - View all

Geometric Theory of Functions of a Complex Variable Gennadiĭ Mikhaĭlovich Goluzin No preview available - 1969 |

### Common terms and phrases

accordance analytic curve analytic functions angle annulus approaches belongs bounded set Cauchy's Chapter circle class of functions closed disk closed Jordan curve coefficients completes the proof conformal mapping Consequently constant contained converges uniformly corresponding defined denote a function equality holding everywhere exists extremal function extremal problems finite number fn(z follows formula func function f(z Furthermore given Goluzin Green's function harmonic function harmonic measure integral interior point interval Jordan curves Let us denote Let us show limiting values mapping function maps the disk maps the domain meromorphic method multiply connected domains n-connected neighborhood obtain obviously plane point ZQ polynomial positive number prove range of values real axis regular and univalent regular function result Riemann surface satisfy the condition Schwarz lemma segment set of points sharp inequality simplex simply connected domain starlike subset sufficiently small tion transfinite diameter univalent functions univalent mapping z-plane

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### References to this book

Invariant Distances and Metrics in Complex Analysis Marek Jarnicki,Peter Pflug No preview available - 1993 |