Value Distribution Theory

It is well known that solving certain theoretical or practical problems often depends on exploring the behavior of the roots of an equation such as (1) J(z) = a, where J(z) is an entire or meromorphic function and a is a complex value. It is especially important to investigate the number n(r, J = a) of the roots of (1) and their distribution in a disk Izl ~ r, each root being counted with its multiplicity. It was the research on such topics that raised the curtain on the theory of value distribution of entire or meromorphic functions. In the last century, the famous mathematician E. Picard obtained the pathbreaking result: Any non-constant entire function J(z) must take every finite complex value infinitely many times, with at most one excep­ tion. Later, E. Borel, by introducing the concept of the order of an entire function, gave the above result a more precise formulation as follows. An entire function J (z) of order A( 0 < A oo logr for every finite complex value a, with at most one exception. This result, generally known as the Picard-Borel theorem, lay the foundation for the theory of value distribution and since then has been the source of many research papers on this subject. Introduction Contents Chapter 1. Essentials of Nevanlinna Theory 1.1 The Poisson-Jensen Formula 1.1.1 The Poisson-Jensen formula 1.1.2 Corollaries 1.2 Characteristic Functions and the First Fundamental Theorem 1.2.1 Characteristic function 1.2.2 Characteristic functions of products and sums; Examples 1.2.3 First fundamental theorem; Properties of characteristic functions 1.3 The Second Fundamental Theorem 1.3.1 A simple form of the second fundamental theorem 1.3.2 A general form of the second fundamental theorem 1.3.3 The fundamental lemma on the logarithmic derivative 1.3.4 The Borel Lemma 1.3.5 Error terms in the second fundamental theorem 1.4 Applications of the Second Fundamental Theorem 1.4.1 The Picard-Borel theorem 1.4.2 Deficiency relation 1.5 Generalizations of the Second Fundamental Theorem 1.5.1 A special result 1.5.2 Proof of Theorem 1.13 Chapter 2. Normal Families 2.1 Normal Families of Holomorphic Functions 2.1.1 Definition and fundamental properties 2.1.2 Criteria for normality 2.1.3 Marty criterion 2.2 Montel's Criterion 2.2.1 Preliminary Lemmas 2.2.2 Theorems of bounded type 2.2.3 Montel's criterion 2.3 Montel Cycle, Normal Families of Meromorphic Functions 2.3.1 Montel cycle 2.3.2 Normal families of meromorphic functions Chapter 3. Borel Directions 3.1 Preliminaries 3.1.1 Boutroux-Cartan theorem 3.1.2 Spherical distance 3.1.3 Theorem of bounded type 3.2 Fundamental Theorems 3.2.1 Valiron's fundamental theorem 3.2.2 Generalizations 3.2.3 Rauch Theorem 3.3 Filling Disks and Borel Directions 3.3.1 Two lemmas 3.3.2. Filling disks and Julia directions of a meromorphic function 3.3.3 Filling disks and Borel directions of order A of a meromorphic function 3.4 Properties of Borel Directions 3.4.1 One lemma 3.4.2 Properties of meromorphic functions in an angle 3.4.3 A sequence of filling disks determined by a Borel direction Chapter 4. Value Distribution of Meromorphic Functions Together with Their Derivatives 4.1 Comparison Between Growths of T(r, f) and T(r, f') 4.1.1 Two lemmas 4.1.2 Comparison between T(r, f) and T(r, f') 4.2 Modular Distribution of Meromorphic Functions Together with Their Derivatives 4.2.1 Generalization of the Nevanlinna fundamental lemma on logarithmic derivative 4.2.2 An inequality of Milloux 4.2.3 An inequality of Hiong King-lai 4.3 An Inequality of Hayman 4.3.1 A lemma of Hayman 4.3.2 The Hayman inequality 4.4 A General Criterion for Normality 4.4.1 Preliminary lemma 4.4.2 A general criterion for normality 4.4.3 Hayman directions 4.5 Total Deficiency of Meromorphic Derivatives 4.5.1 A lemma of Frank and Weissenborn 4.5.2 Precise estimate of the total deficiency of meromorphic derivatives 4.5.3 Problems of Drasin 4.5.4 Conjectures of Frank, Goldberg and Mues 4.6 A New Method for Proving Normality 4.6.1 A lemma of Zalcman 4.6.2 New method for proving normality Chapter 5. Recent Studies on Borel Directions 5.1 Distribution of Borel Directions 5.1.1 Preliminary lemmas 5.1.2 Distribution of Borel directions 5.2 Common Borel Directions of a Meromorphic Function and Its Derivatives 5.2.1 On the Milloux theorem 5.2.2 Fundamental theorem 5.2.3 Corollaries 5.3 Angular Distribution of Meromorphic Functions Together with Their Derivatives 5.3.1 Preliminary lemma 5.3.2 Fundamental theorem 5.3.3 New singular directions Chapter 6. Deficient Values and Borel Directions of Meromorphic Functions 6.1 Precise Order and Three Lemmas 6.1.1 Precise order 6.1.2. Three lemmas 6.2 Distribution of Borel Directions of Meromorphic Functions with Deficient Values 6.2.1. Main result 6.2.2 Discussion 6.3 Deficient Values and Borel Directions of Meromorphic Functions 6.3.1 Deficient values and Borel directions 6.3.2 Complement 6.4 Deficient Values, Borel Directions and the Order of Entire Functions 6.4.1 Several lemmas 6.4.2 Deficient values and Borel directions of an entire function 6.4.3 Number of deficient values and order of an entire function Chapter 7. The Spread Relation and Its Applications 7.1 Sequence of Pólya Peaks and Its Existence 7.1.1 Definition and lemmas 7.1.2 Existence of sequence of Pólya peaks 7.2 The T* Function 7.2.1 Definition and continuity of T* function 7.2.2 Subharmonicity of the T* function 7.3 The Spread Relation 7.3.1 Definition 7.3.2 Spread relation 7.4 Applications of the Spread Relation 7.4.1 Fuchs' theorem 7.4.2 Ellipse theorem 7.5 The Deficiency Problem 7.5.1 The deficiency problem 7.5.2 A problem in convex programming 7.5.3 Settlement of the deficiency problem in the case of μ < 1 Bibliography A-B B-D D-F F-G G-H H-M M-S S-W W-Y Y-Z Index It is well known that solving certain theoretical or practical problems often depends on exploring the behavior of the roots of an equation such as (1) J(z) = a, where J(z) is an entire or meromorphic function and a is a complex value. It is especially important to investigate the number n(r, J = a) of the roots of (1) and their distribution in a disk Izl ~ r, each root being counted with its multiplicity. It was the research on such topics that raised the curtain on the theory of value distribution of entire or meromorphic functions. In the last century, the famous mathematician E. Picard obtained the pathbreaking result: Any non-constant entire function J(z) must take every finite complex value infinitely many times, with at most one excepƯ tion. Later, E. Borel, by introducing the concept of the order of an entire function, gave the above result a more precise formulation as follows. An entire function J (z) of order A(0