Theory of Entire and Meromorphic Functions--Deficient and Asymptotic Values and Singular Directions

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Translations of Mathematical Monographs 122

S Title

Theory of Entire and Meromorphic Functions: Deficient and AsymptoticValues and Singular Directions

Copyright (C) 1993 by the American Mathematical Society
     ISBN 0-8218-4589-6
     QA353.E5Z4313 1993 515'.98-dc20
     93-43 CIP

Contents

Preface

CHAPTER 1  The Nevanlinna Theory
     §1.1. The Poisson-Jensen formula
          1.1.1. The Poisson-Jensen formula
          1.1.2. The Jensen-Nevanlinna formula
     § 1.2. The characteristic function
          1.2.1. Definition of the characteristic function.
          1.2.2. Cartan's identical relation
          1.2.3. Some inequalities of the characteristic function
          1.2.4. The relationships between the maximum modulus and the characteristic function of a regular function
     §1.3. The Ahlfors-Shimizu characteristic
          1.3.1. The Ahlfors-Shimizu characterist
          1.3.2. The relationship between T(r, f) and To(r, f).
     § 1.4. The First Fundamental Theorem
          1.4.1. The First Fundamental Theorem
          1.4.2. Growth of a transcendental meromorphic function.
          1.4.3. Examples
          1.4.4. Orders
     §1.5. Lemma on the logarithmic derivative
          1.5.1. Two lemmas
          1.5.2. Lemma on the logarithmic derivative
          1.5.3. Borel Lemma
     §1.6. The Second Fundamental Theorem
          1.6.1. The Second Fundamental Theorem
          1.6.2. Applications
     § 1.7. Annotated notes
          1.7.1. Milloux's inequality
          1.7.2. Hayman's inequality
          1.7.3. Zhuang Chi-tai's inequality.
          1.7.4. The Ahlfors Theory

CHAPTER 2  The Singular Directions
     §2.1. On some properties of monotonic functions
     §2.2. The Boutroux-Cartan Theorem
          2.2.1. The Boutroux-Cartan Theorem
          2.2.2. Extensions
     §2.3. Fundamental theorem of value distribution of functions meromorphic in a disk
          2.3.2. The theorem of bound
          2.3.2. The fundamental theorem
          2.3.3. Schottky-type theorem
     §2.4. The Julia and Borel directions
          2.4.1. The filling circle
          2.4.2. The Borel direction
          2.4.3. The Julia direction
     §2.5. On the growth of the entire function
          2.5.1. Some lemmas
          2.5.2. Distribution of the Julia directions
     §2.6. On the Nevanlinna direction
          2.6.1. Definition of the Nevanlinna direction
          2.6.2. Some lemmas.
          2.6.3. Theorem on the existence of the Nevanlinna direction
     §2.7. Annotated notes
          2.7.1. The common Borel direction
          2.7.2. The distribution regularity of the Borel direction

CHAPTER 3  The Deficient Value Theory
     §3.1. The harmonic measure and the Lindelof-type theorem
          3.1.1. An estimation on the harmonic measure
          3.1.2. A local version of the Lindelof Theorem
     §3.2. The Length-Area Principle
          3.2.1. The Length-Area Principle
          3.2.2. Applications.
     §3.3. On the growth of meromorphic functions with deficient values
          3.3.1. Growth of a meromorphic function and its deficient values
          3.3.2. A lemma about the deficient values
          3.3.3. On the growth and distribution of zeros and poles of meromorphic functions
     §3.4. The Weitsman Theorem
     §3.5. The Edrei-Fuchs Theorem
          3.5.1. Some preparations
          3.5.2. The Edrei-Fuchs Theorem
          3.5.3. Improvement of the Edrei-Fuchs Theorem
     §3.6. Annotated notes
          3.6.1. Inverse problem
          3.6.2. Spread relation
          3.6.3. F. Nevanlinna's Conjecture

CHAPTER 4  The Asymptotic Value Theory
     §4.1. The asymptotic value and the transcendental singularity
          4.1.1. The fundamental concept [32b, 39a]
          4.1.2. The Iversen Theorem
          4.1.3. The Lindelof Theorem
          4.1.4. The Fuchs Theorem.
     §4.2. The Denjoy Conjecture
          4.2.1. The Denjoy Conjecture
          4.2.2. Entire functions that satisfy the extreme case k = 2A of the Denjoy Conjecture.
     §4.3. Growth of entire functions along an asymptotic path
     §4.4. An estimate on the length of the asymptotic path of an entire function
     §4.5. Direct transcendental singularities
          4.5.1. The Ahlfors Theorem.
          4.5.2. Two lemmas
          4.5.3. The distribution of zeros and poles of a meromorphic function and the direct transcendental singularities of its inverse function.

CHAPTER 5  The Relationship between Deficient Values and Asymptotic Values of an Entire Function
     §5.1. The theorem of the bound and its application regarding functions meromorphic in the unit disk
          5.1.1. The theorem of the bound
          5.1.2. Applications
     §5.2. Entire functions of finite lower order [43c]
     §5.3. On entire functions having a finite number of Julia directions [43h]
     §5.4. Extremal length and Ahlfors Distortion Theorem
          5.4.1. Extremal length
          5.4.2. Rule of composition and the symmetry principle
          5.4.3. Two problems on extremals
          5.4.4. Ahlfors' Distortion Theorem
     §5.5. On entire functions with zeros distributed on a finite number of half lines [43j]

CHAPTER 6  The Relationship between Deficient Values of a Meromorphic Function and Direct Transcendental Singularities of its Inverse Functions
     §6.1. On meromorphic functions having deficiency sum two [43g]
     §6.2. On meromorphic functions of finite lower order [43c]

Some Supplementary Results

References

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