Univalent Functions

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Springer Science & Business Media, Jul 2, 2001 - Mathematics - 384 pages
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The theory of univalent functions is a fascinating interplay of geometry and analysis, directed primarily toward extremal problems. A branch of complex analysis with classical roots, it is an active field of modern research. This book describes the major
 

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Contents

Geometric Function Theory
1
12 Local Mapping Properties
5
13Normal Families
7
14 Extremal Problems
10
15 The Riemann Mapping Theorem
11
16 Analytic Continuation
12
17 Harmonic and Subharmonic Functions
15
18 Greens Functions
19
510 Successive Coefficients of Starlike Functions
177
511 Exponentiation of the Goluzin Inequalities
180
512 FitzGeralds Theorem
183
Exercise
187
Subordination
190
62 Coefficient Inequalities
192
63 Sharpened Forms of the Schwarz Lemma
197
64 Majorization
202

19 Positive Harmonic Functions
21
Excercise
24
Elementary Theory of Univalent Functions
26
22 The Area Theorem The univalence of a function
29
23 Growth and Distortion Theorems
32
24 Coefficient Estimates We have seen that each function
36
25 Convex and Starlike Functions
40
26 ClosetoConvex Functions
46
27 Spirallike Functions
52
28 Typically Real Functions
55
29 A Primitive Variational Method
58
210 Growth of Integral Means
60
211 Odd Univalent Functions
64
212 Asymptotic Bieberbach Conjecture
66
Notes
69
Exercise
70
Parametric Representation of Slit Mappings
76
32 Density of Slit Mappings
80
33 Loewners Differential Equation
82
34 Univalence of Solutions
87
35 The Third Coefficient
93
36 Radius of Starlikeness
95
37 The Rotation Theorem
98
38 Coefficients of Odd Functions
103
39 An Elementary Counterexample
107
310 Robertsons Conjecture
110
311 Successive Coefficients
113
Exercise
115
Generalizations of the Area Principle
118
42 Polynomial Area Theorem
120
43 The Grunsky Inequalities
122
44 Inequalities of Goluzin and Lebedev
125
45 Unitary Matrices
128
46 The Fourth Coefficient
131
47 Coefficient Problem in the Class
134
Notes
139
Exercise
140
Exponentiation of the Grunsky Inequalities
142
52 Reformulation of the Grunsky Inequalities
146
53 Estimation of the nth Coefficient
149
54 Logarithmic Coefficients
151
55 Radial Growth
157
56 Bazilevichs Theorem
159
57 Haymans Regularity Theorem
162
58 Proof of Milins Tauberian Theorem
168
59 Successive Coefficients
172
65 Univalent Subordinate Functions
207
Exercise
212
Integral Means
214
72 The StarFunction
216
73 Proof of Baernsteins Theorem
219
74 Subharmonic Property of the StarFunction
225
75 Integral Means of Derivatives
229
Exercise
232
Some Special Topics
234
82 Sections of Univalent Functions
243
83 Convolutions of Convex Functions
246
84 Coefficient Multipliers
254
85 Criteria for Univalence
258
86 Additional Topics
265
2 Univalent Polynomials Which polynomials of the form
267
3 Functions of Bounded Boundary Rotation
269
Exercise
271
General Extremal Problems
275
92 Representation of Linear Functionals
278
93 Extreme Points and Support Points
280
94 Properties of Extremal Functions
283
95 Extreme Points of S
286
96 Extreme Points of Z
288
Exercise
290
Boundary Variation
292
102 Conformal Radius
293
103 Schiffers Theorem
295
104 Local Structure of Trajectories
302
105 Application to Extremal Problems
304
106 Support Points of S
306
107 PointEvaluation Functionals
314
108 The Coefficient Problem
318
109 Region of Values of log CC
323
1010 Multiply Connected Domains
326
1011 Other Variational Methods
328
Exercise
330
Coefficient Regions
334
112 Boundary Points
338
113 Canonical Differential Equation
343
114 Algebraic Functions
346
Exercise
352
Suggestions for Further Reading
355
Bibliography
357
List of Symbols
377
Index
379
Copyright

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Page 362 - JA Hummel, A variational method for starlike functions, Proc. Amer. Math. Soc., 9 (1958), 82-87.
Page 371 - TECHNISCHE HOCHSCHULE, BRAUNSCHWEIG, GERMANY. THE COEFFICIENT PROBLEM FOR SCHLICHT MAPPINGS OF THE EXTERIOR OF THE UNIT CIRCLE GEORGE SPRINGER Let E(q) represent the domain consisting of the whole...
Page 361 - Some new properties of support points for compact families of univalent functions in the unit disc.

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