Table of Contents
Preface v
List of Symbols xi
1 Univalent Functions - the Elementary Theory 1
1.1 Definitions and Basic Properties 1
1.2 Bieberbach's Conjecture and Related Topics 3
1.3 Growth and Distortion Theorems 8
2 Definitions of Major Subclasses 13
2.1 Convex and Starlike Functions 13
2.2 Close-to-Convex Functions 16
23 Bazilevic Functions 18
2.4 The Class U 20
2.5 Rotational Invariance 20
3 Fundamental Lemmas 22
3.1 Functions with Positive Real Part 22
3.2 Subordination 31
3.3 The Clunie-Jack Lemma 35
4 Starlike and Convex Functions 37
4.1 Starlike Functions 37
4.1.1 Coefficient Theorems 37
4.1.2 Refined Growth Theorems 42
4.13 Theorems Concerning lim $$$ arg f(reiθ) 45
4.1.4 The Radial Limit lim $$$ f(reiθ) 47
4.1.5 Length and Integral Mean Problems 50
4.1.6 Some Subclasses of Starlike Functions 54
4.2 Convex Functions 58
4.2.1 Growth and Distortion Theorems 59
4.2.2 Coefficient Inequalities 60
5 Starlike and Convex Functions of Order α 65
5.1 Definitions and Growth and Distortion Theorems 65
5.2 Inclusion Relationships 67
5.3 Coefficient Theorems 75
5.4 Sufficient Conditions 78
5.4.1 Sufficient Conditions on ″ (z) 78
5.4.2 On a Class Defined by Silverman 81
6 Strongly Starlike and Convex Functions 83
6.1 Definitions 83
6.2 Strongly Starlike Functions 83
6.3 Coefficient Theorems 87
6.4 Strongly Convex Functions 93
6.5 Inclusion Relationships 95
7 Alpha-Convex Functions 99
7.1 Definition and Integral Representation 99
7.2 Distortion and Growth Theorems 101
7.3 Coefficient Problems 106
8 Gamma-Starlike Functions 112
8.1 Definition and Basic Properties 112
8.2 Coefficient Inequalities 113
8.2.1 Logarithmic Coefficients 116
8.2.2 Inverse Coefficients 118
8.2.3 The Second Hankel Determinant 119
9 Close-to-Convex Functions 121
9.1 Definitions and Basic Properties 121
9.2 Distortion Theorems 124
9.3 Coefficient Problems 125
9.3.1 The Fekete-Szego Problem 125
9.3.2 The Zalcman Conjecture 127
9.3.3 Difference of Coefficients 132
9.3.4 Robertson's Conjecture 134
9.3.5 Logarithmic Coefficients 136
9.3.6 The Second Hankel Determinant 143
9.4 Growth Estimates 143
9.5 Ozaki Close-to-Convex functions 149
9.5.1 Growth and Area Estimates 151
9.5.2 Strongly Ozaki Close-to-Convex Functions 151
10 Bazilevic Functions 153
10.1 Definition and Basic Properties 153
10.2 Growth Theorems 154
10.2.1 Coefficients of Powers of Functions in B(α) 159
10.2.2 Logarithmic Coefficients 162
10.3 The Fekete-Szego Problem 162
10.4 Sufficient Conditions for f ∈ B(α) 163
11 B1 (α) Bazilevic Functions 165
11.1 Definition and Basic Properties 165
11.2 Distortion Theorems 165
11.3 Growth Estimates 169
11.4 Coefficients 171
11.5 Other Inequalities 174
12 The Class U(λ) 179
12.1 Definition and Geometrical Properties 179
12.2 Sufficient Conditions and Univalence 182
12.3 Coefficients 188
13 Convolutions 195
13.1 Definition and the Pólya-Schoenberg Conjecture 195
13.2 Subordination and Convolution 201
14 Meromorphic Univalent Functions 205
14.1 The Class ∑ 205
14.1.1 Coefficients and the Clunie Constant 205
14.1.2 Coefficients of the Inverse Function 209
14.1.3 Distortion Theorems 209
14.2 Subclasses of ∑ 210
14.2.1 Meromorphic Starlike Functions 210
14.2.2 Meromorphic Close-to-Convex Functions 211
14.2.3 Meromorphic Bazilevic Functions 213
15 Loewner Theory 215
15.1 The Loewner Equation 215
15.2 Applications 216
16 Other Topics 224
16.1 Harmonic Univalent Functions 224
16.2 Bi-univalent Functions 225
16.3 Functions of Bounded Boundary Rotation 226
16.4 Differential Subordinations 229
16.5 Operators 230
16.5.1 The Salagean Operator 230
16.5.2 The Libera Operator and Generalizations 232
17 Open Problems 235
Concluding Remarks 238
Bibliography 239
Index 250
