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1 Preliminaries of Real Functions 1

1.1 Functions of a Real Variable 1

1.1.1 The Order and Lower Order of a Real Function 1

1.1.2 The Pólya Peak Sequence of a Real Function 4

1.1.3 The Regularity of a Real Function 8

1.1.4 Quasi-invariance of Inequalities 14

1.2 Integral Formula and Integral Inequalities 19

1.2.1 The Green Formula for Functions with Two Real Variables 19

1.2.2 Several Integral Inequalities 20

References 23

2 Characteristics of a Meromorphic Function 25

2.1 Nevanlinna's Characteristic in a Domain 26

2.2 Nevanlinna's Characteristic in an Angle 47

2.3 Tsuji's Characteristic 58

2.4 Ahlfors-Shimizu's Characteristic 66

2.5 Estimates of the Error Terms 77

2.6 Characteristic of Derivative of a Meromorphic Function 86

2.7 Meromorphic Functions in an Angular Domain 93

2.8 Deficiency and Deficient Values 103

2.9 Uniqueness of Meromorphic Functions Related to Some Angular Domains 110

References 120

3 T Directions of a Meromorphic Function 123

3.1 Notation and Existence of T Directions 124

3.2 T Directions Dealing with Small Functions 130

3.3 Connection Among T Directions and Other Directions 134

3.4 Singular Directions Dealing with Derivatives 146

3.5 The Common T Directions of a Meromorphic Function and Its Derivatives 151

3.6 Distribution of the Julia,Borel Directions and T Directions 163

3.7 Singular Directions of Meromorphic Solutions of Some Equations 166

3.8 Value Distribution of Algebroid Functions 178

References 181

4 Argument Distribution and Deficient Values 185

4.1 Deficient Values and T Directions 186

4.2 Retrospection 202

References 205

5 Meromorphic Functions with Radially Distributed Values 207

5.1 Growth of Such Meromorphic Functions 208

5.2 Growth of Such Meromorphic Functions with Finite Lower Order 215

5.3 Retrospection 222

References 228

6 Singular Values of Meromorphic Functions 229

6.1 Riemann Surfaces and Singularities 230

6.2 Density of Singularities 241

6.3 Meromorphic Functions of Bounded Type 252

References 265

7 The Potential Theory in Value Distribution 267

7.1 Signed Measure and Distributions 268

7.2 δ-Subharmonic Functions 270

7.2.1 Basic Results Concerning δ-Subharrnonic Functions 271

7.2.2 Normality of Family of δ-Subharmonic Functions 275

7.2.3 The Nevanlinna Theory of δ-Subharmonic Functions 285

7.3 Eremenko's Proof of the Nevanlinna Conjecture 292

References 305

Index 307