Geometric Theory of Functions of a Complex Variable

Front Cover
American Mathematical Soc., 1969 - 676 pages
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Contents

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
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Normal Families
Joel L. Schiff
Limited preview - 1993
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