Univalent Functions

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Springer Science & Business Media, Jul 2, 2001 - Mathematics - 384 pages
 

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Contents

Geometric Function Theory
xiii
12 Local Mapping Properties
3
13Normal Families
5
14 Extremal Problems
8
15 The Riemann Mapping Theorem
9
16 Analytic Continuation
10
17 Harmonic and Subharmonic Functions
13
18 Greens Functions
17
510 Successive Coefficients of Starlike Functions
175
511 Exponentiation of the Goluzin Inequalities
178
512 FitzGeralds Theorem
181
Exercise
185
Subordination
188
62 Coefficient Inequalities
190
63 Sharpened Forms of the Schwarz Lemma
195
64 Majorization
200

19 Positive Harmonic Functions
19
Excercise
22
Elementary Theory of Univalent Functions
24
22 The Area Theorem The univalence of a function
27
23 Growth and Distortion Theorems
30
24 Coefficient Estimates We have seen that each function
34
25 Convex and Starlike Functions
38
26 ClosetoConvex Functions
44
27 Spirallike Functions
50
28 Typically Real Functions
53
29 A Primitive Variational Method
56
210 Growth of Integral Means
58
211 Odd Univalent Functions
62
212 Asymptotic Bieberbach Conjecture
64
Notes
67
Exercise
68
Parametric Representation of Slit Mappings
74
32 Density of Slit Mappings
78
33 Loewners Differential Equation
80
34 Univalence of Solutions
85
35 The Third Coefficient
91
36 Radius of Starlikeness
93
37 The Rotation Theorem
96
38 Coefficients of Odd Functions
101
39 An Elementary Counterexample
105
310 Robertsons Conjecture
108
311 Successive Coefficients
111
Exercise
113
Generalizations of the Area Principle
116
42 Polynomial Area Theorem
118
43 The Grunsky Inequalities
120
44 Inequalities of Goluzin and Lebedev
123
45 Unitary Matrices
126
46 The Fourth Coefficient
129
47 Coefficient Problem in the Class
132
Notes
137
Exercise
138
Exponentiation of the Grunsky Inequalities
140
52 Reformulation of the Grunsky Inequalities
144
53 Estimation of the nth Coefficient
147
54 Logarithmic Coefficients
149
55 Radial Growth
155
56 Bazilevichs Theorem
157
57 Haymans Regularity Theorem
160
58 Proof of Milins Tauberian Theorem
166
59 Successive Coefficients
170
65 Univalent Subordinate Functions
205
Exercise
210
Integral Means
212
72 The StarFunction
214
73 Proof of Baernsteins Theorem
217
74 Subharmonic Property of the StarFunction
223
75 Integral Means of Derivatives
227
Exercise
230
Some Special Topics
232
82 Sections of Univalent Functions
241
83 Convolutions of Convex Functions
244
84 Coefficient Multipliers
252
85 Criteria for Univalence
256
86 Additional Topics
263
2 Univalent Polynomials Which polynomials of the form
265
3 Functions of Bounded Boundary Rotation
267
Exercise
269
General Extremal Problems
273
92 Representation of Linear Functionals
276
93 Extreme Points and Support Points
278
94 Properties of Extremal Functions
281
95 Extreme Points of S
284
96 Extreme Points of Z
286
Exercise
288
Boundary Variation
290
102 Conformal Radius
291
103 Schiffers Theorem
293
104 Local Structure of Trajectories
300
105 Application to Extremal Problems
302
106 Support Points of S
304
107 PointEvaluation Functionals
312
108 The Coefficient Problem
316
109 Region of Values of log CC
321
1010 Multiply Connected Domains
324
1011 Other Variational Methods
326
Exercise
328
Coefficient Regions
332
112 Boundary Points
336
113 Canonical Differential Equation
341
114 Algebraic Functions
344
Exercise
350
Suggestions for Further Reading
353
Bibliography
355
List of Symbols
375
Index
377
Copyright

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Page 360 - JA Hummel, A variational method for starlike functions, Proc. Amer. Math. Soc., 9 (1958), 82-87.
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