Value distribution of meromorphic functions.

*(English)*Zbl 1233.30004
Berlin: Springer; Beijing: Tsinghua University Press (ISBN 978-7-302-22329-0/hbk; 978-3-642-12908-7/hbk; 978-3-642-12909-4/ebook). vi, 308 p. (2010).

This is a monograph devoted to the value distribution theory of meromorphic functions, also known as Nevanlinna theory. There are several monographs on Nevanlinna theory, by R. Nevanlinna himself [Eindeutige analytische Funktionen. Berlin: Springer (1936; Zbl 0014.16304; JFM 62.0315.02)], by W. Hayman [Meromorphic functions. Oxford Mathematical Monographs. Oxford: At the Clarendon Press (1964; Zbl 0115.06203)], M. Tsuji [Potential theory in modern function theory. Tokyo: Maruzen Co., Ltd. (1959; Zbl 0087.28401)] and others. The most complete exposition of the theory can be found in the book [Value distribution of meromorphic functions (Russian). Moscow: Nauka (1970; Zbl 0217.10002)] by A. A. Gol’dberg and I. V. Ostrovskii. This book has been translated into English and published by the AMS in 2008 [A. A. Goldberg and I. V. Ostrovskii, Value distribution of meromorphic functions. Translations of Mathematical Monographs 236. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1152.30026)], and it contains a survey by A. Eremenko and J. K. Langley of the main results in the theory after 1970. However, most of them concern meromorphic functions in the plane.

The book under review complements the mentioned monographs. Each chapter finishes with a list of references (in total there are more than 200 items) containing both classical and recent publications on the topic, not only in English but also in Chinese and Russian.

Besides the exposition of the background of Nevanlinna theory in Chapter 2, the material of the monograph is related to the author’s own investigations. The results mostly concern functions meromorphic in a closed angle, and they are concentrated near the concept of the \(T\) direction (Chapters 3-5). Let \(N_\varepsilon(r,a, f)\) be the Nevanlinna counting function of \(a\)-points of \(f\) in the sector with opening \(\varepsilon\) and a direction \(L_\theta=\{\arg z=\theta\}\) as the bisector. The direction \(L_\theta\) is called a \(T\) direction of a meromorphic function \(f\) in an angular domain \(\Omega\supset L_\theta\) if \(\varlimsup_{r\to \infty} \frac{N_\varepsilon(r,a,f)}{T(r,f)}>0\) for every \(\varepsilon>0\) and all \(a\in \hat{\mathbb C}\), with possibly two exceptions, where \(T(r,f)\) is the Ahlfors-Shimizu characteristic of \(f\) for \(\Omega\).

The last chapter contains, in particular, basics of Nevanlinna theory for \(\delta\)-subharmonic functions and Eremenko’s proof of F. Nevanlinna’s conjecture (in terms of potential theory), originally proved by D. Drasin [“Proof of a conjecture of F. Nevanlinna concerning functions which have deficiency sum two”, Acta Math. 158, 1–94 (1987; Zbl 0622.30028)].

The following is an outline of the contents of the seven chapters:

1. Preliminaries of real functions: orders, Polýa’s peaks, densities, Borel-Nevanlinna type lemmas;

2. Characteristics of a meromorphic function: Poisson-Jensen formula, Nevanlinna’s characteristic for a domain, the fundamental theorems of R. Nevanlinna, Boutroux-Cartan’s lemma, Tsuji’s characteristic, Ahlfors-Shimizu’s characteristic, logarithmic derivative estimates, deficiencies and deficient values, spread relation, Borel’s direction, uniqueness theorems;

3. \(T\) directions of a meromorphic function: \(T\) direction, Julia’s direction, Borel’s direction, Hayman’s direction;

4. Argument distribution and deficient values: comparison of numbers of deficient values and singular directions;

5. Meromorphic functions with radially distributed values: growth of meromorphic functions with radially distributed \(a\)-values;

6. Singular values of meromorphic functions: Riemann surfaces, singularities, asymptotic values, critical values, meromorphic functions of bounded type;

7. The potential theory in value distribution: \(\delta\)-subharmonic functions, Grishin’s lemma, normal families of \(\delta\)-subharmonic functions, convergence, Zalcman’s lemma, Nevanlinna theory for \(\delta\)-subharmonic functions, Eremenko’s proof of F. Nevanlinna’s conjecture.

I would recommend this book to any person who is interested in value distribution theory in an angular domain. However, the case when \(f\) is meromorphic in an open angle is omitted here. We address the reader to results of [N. V. Govorov, Kraevaya zadacha Rimana s beskonechnym indeksom (Russian). Moskva: Nauka (1986; Zbl 0632.30001)] on asymptotic properties of analytic functions in an angle, and the paper by M. A. Fedorov and A. F. Grishin [Math. Phys. Anal. Geom. 1, No. 3, 223–271 (1998, Zbl 0913.30020)]. I would add to Chapter 1 a non-trivial generalization of Pólya’s peaks for functions infinite order obtained by W. Bergweiler in [Analysis 10, No. 2–3, 163–176 (1990; Zbl 0703.30025)].

The book under review complements the mentioned monographs. Each chapter finishes with a list of references (in total there are more than 200 items) containing both classical and recent publications on the topic, not only in English but also in Chinese and Russian.

Besides the exposition of the background of Nevanlinna theory in Chapter 2, the material of the monograph is related to the author’s own investigations. The results mostly concern functions meromorphic in a closed angle, and they are concentrated near the concept of the \(T\) direction (Chapters 3-5). Let \(N_\varepsilon(r,a, f)\) be the Nevanlinna counting function of \(a\)-points of \(f\) in the sector with opening \(\varepsilon\) and a direction \(L_\theta=\{\arg z=\theta\}\) as the bisector. The direction \(L_\theta\) is called a \(T\) direction of a meromorphic function \(f\) in an angular domain \(\Omega\supset L_\theta\) if \(\varlimsup_{r\to \infty} \frac{N_\varepsilon(r,a,f)}{T(r,f)}>0\) for every \(\varepsilon>0\) and all \(a\in \hat{\mathbb C}\), with possibly two exceptions, where \(T(r,f)\) is the Ahlfors-Shimizu characteristic of \(f\) for \(\Omega\).

The last chapter contains, in particular, basics of Nevanlinna theory for \(\delta\)-subharmonic functions and Eremenko’s proof of F. Nevanlinna’s conjecture (in terms of potential theory), originally proved by D. Drasin [“Proof of a conjecture of F. Nevanlinna concerning functions which have deficiency sum two”, Acta Math. 158, 1–94 (1987; Zbl 0622.30028)].

The following is an outline of the contents of the seven chapters:

1. Preliminaries of real functions: orders, Polýa’s peaks, densities, Borel-Nevanlinna type lemmas;

2. Characteristics of a meromorphic function: Poisson-Jensen formula, Nevanlinna’s characteristic for a domain, the fundamental theorems of R. Nevanlinna, Boutroux-Cartan’s lemma, Tsuji’s characteristic, Ahlfors-Shimizu’s characteristic, logarithmic derivative estimates, deficiencies and deficient values, spread relation, Borel’s direction, uniqueness theorems;

3. \(T\) directions of a meromorphic function: \(T\) direction, Julia’s direction, Borel’s direction, Hayman’s direction;

4. Argument distribution and deficient values: comparison of numbers of deficient values and singular directions;

5. Meromorphic functions with radially distributed values: growth of meromorphic functions with radially distributed \(a\)-values;

6. Singular values of meromorphic functions: Riemann surfaces, singularities, asymptotic values, critical values, meromorphic functions of bounded type;

7. The potential theory in value distribution: \(\delta\)-subharmonic functions, Grishin’s lemma, normal families of \(\delta\)-subharmonic functions, convergence, Zalcman’s lemma, Nevanlinna theory for \(\delta\)-subharmonic functions, Eremenko’s proof of F. Nevanlinna’s conjecture.

I would recommend this book to any person who is interested in value distribution theory in an angular domain. However, the case when \(f\) is meromorphic in an open angle is omitted here. We address the reader to results of [N. V. Govorov, Kraevaya zadacha Rimana s beskonechnym indeksom (Russian). Moskva: Nauka (1986; Zbl 0632.30001)] on asymptotic properties of analytic functions in an angle, and the paper by M. A. Fedorov and A. F. Grishin [Math. Phys. Anal. Geom. 1, No. 3, 223–271 (1998, Zbl 0913.30020)]. I would add to Chapter 1 a non-trivial generalization of Pólya’s peaks for functions infinite order obtained by W. Bergweiler in [Analysis 10, No. 2–3, 163–176 (1990; Zbl 0703.30025)].

Reviewer: Igor Chyzhykov (Lviv)

##### MSC:

30-02 | Research exposition (monographs, survey articles) pertaining to functions of a complex variable |

30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |

31A05 | Harmonic, subharmonic, superharmonic functions in two dimensions |