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**Uniqueness theory of meromorphic functions.**
*(English)*
Zbl 1070.30011

Mathematics and its Applications (Dordrecht) 557. Dordrecht: Kluwer Academic Publishers. Beijing: Science Press (ISBN 1-4020-1448-1/hbk; 7-03-008451-9/hbk). viii, 569 p. (2003).

The uniqueness theory of transcendental meromorphic functions goes back to R. Nevanlinna who proved that any non-constant meromorphic function can be determined by five values applying the value distribution theory established by himself. The uniqueness theory of entire and meromorphic functions mainly studies conditions under which there exists essentially only one function satisfying these conditions. There has been considerable research concerning the uniqueness theory under various different situations. This book is the first exposition systematically summarizing recent results, and also presenting useful skills in this field.

This book is constructed in 10 chapters, 42 sections and 182 subsections. It starts with the introduction of the Nevanlinna theory. This contains the Valiron-Mokhon’ko theorem, the Frank-Weissenborn theorem, the Steinmetz theorem for small functions, Borel’s identities and the Niino theorem which play important roles in this field. Chapter 2 deals with uniqueness results for finite order functions. It contains Anastassiadis’ works relating to the coefficients of Taylor expansions of some entire functions of finite order. Chapter 3 discusses the Nevanlinna five values theorem and contains various improvements and generalizations of it, for example L. Yang’s method for multiple values and Zhang’s idea for small functions. Unicity theory of algebroid function due to Y. Z. He is introduced. The next two chapters are perhaps main parts of this book. In Chapter 4, the authors collects various improvements and generalizations of the Nevanlinna four values theorem, including Gundersen’s \(2\text{CM}+ 2\text{IM}= 4\text{CM}\) theorem, and Mues’ and Ueda’s works. They also give Gundersen’s example for 4DM case, and Reinders’ 4DM values theorem. Chapter 5 is devoted to the studies with the condition that functions share three values. It contains Yi’s works, and Brosch’s work related to uniqueness of periodic functions. Uniqueness of the solutions of differential equation is also discussed. Chapter 6 is concerned with three value sets of meromorphic functions, containing Ozawa’s work and series of Ueda’s works. Chapter 7 contains various kinds of two values theorems and also one value theorems. In the next two chapters, they treat sharing value problems on meromorphic functions and their derivatives. Chapter 8 contains Frank’s work, and Mues and Reinders’ result on functions having two CM or three IM shared values with their differential polynomials. Chapter 9 is devoted to the studies of Hinkkanen’s problem, see [K. F. Barth, D. A. Brannan and W. K Hayman, Research problems in complex analysis, Bull. Lond. Math. Soc. 16, 490–517 (1984; Zbl 0593.30001)]. Köhler and Tohge’s results are introduced. They also discuss C. C. Yang’s problem on the situation that functions share one value and their derivatives share one value. Chapter 10 introduces research works on the unique range set of meromorphic functions which contain results on meromorphic functions sharing four, three, or two sets. In particular, Gross-Yang’s problem is discussed.

A lot of research papers of uniqueness theory of meromorphic functions appeared recently. Summarizing these results may be a great work, and it may serve as a useful guide to the progress of recent years. I add recent two topics in uniqueness theory of meromorphic functions, one being the studies between this theory and a normal family, see e.g. [M. Fang and L. Zalcman, A note on normality and shared values, J. Aust. Math. Soc. 76, No. 1, 141–150 (2004; Zbl 1074.30032)], another one is the studies with some conditions in angular domains, see, e.g. [J.-H. Zheng, On uniqueness of meromorphic functions with shared values in some angular domains, Can. Math. Bull. 47, No. 1, 152–160 (2004; Zbl 1045.30019)].

Finally, I mention some of most recent results relating with some contents in this book. By Yamanoi, the second fundamental theorem for small functions is obtained, which may give short proofs discussed in Chapter 1, see [K. Yamanoi, The second main theorem for small functions and related problems, Acta Math. 192, No. 1, 225–294 (2004; Zbl 1203.30035)]. In connection with Chapter 9, Hinkkanen’s problem is solved by G. Frank, X. Hua and R. Vaillancourt, see [Meromorphic functions sharing the same zeros and poles, Can. J. Math. 56, No. 6, 1190–1227 (2004; Zbl 1065.30027)].

This book is constructed in 10 chapters, 42 sections and 182 subsections. It starts with the introduction of the Nevanlinna theory. This contains the Valiron-Mokhon’ko theorem, the Frank-Weissenborn theorem, the Steinmetz theorem for small functions, Borel’s identities and the Niino theorem which play important roles in this field. Chapter 2 deals with uniqueness results for finite order functions. It contains Anastassiadis’ works relating to the coefficients of Taylor expansions of some entire functions of finite order. Chapter 3 discusses the Nevanlinna five values theorem and contains various improvements and generalizations of it, for example L. Yang’s method for multiple values and Zhang’s idea for small functions. Unicity theory of algebroid function due to Y. Z. He is introduced. The next two chapters are perhaps main parts of this book. In Chapter 4, the authors collects various improvements and generalizations of the Nevanlinna four values theorem, including Gundersen’s \(2\text{CM}+ 2\text{IM}= 4\text{CM}\) theorem, and Mues’ and Ueda’s works. They also give Gundersen’s example for 4DM case, and Reinders’ 4DM values theorem. Chapter 5 is devoted to the studies with the condition that functions share three values. It contains Yi’s works, and Brosch’s work related to uniqueness of periodic functions. Uniqueness of the solutions of differential equation is also discussed. Chapter 6 is concerned with three value sets of meromorphic functions, containing Ozawa’s work and series of Ueda’s works. Chapter 7 contains various kinds of two values theorems and also one value theorems. In the next two chapters, they treat sharing value problems on meromorphic functions and their derivatives. Chapter 8 contains Frank’s work, and Mues and Reinders’ result on functions having two CM or three IM shared values with their differential polynomials. Chapter 9 is devoted to the studies of Hinkkanen’s problem, see [K. F. Barth, D. A. Brannan and W. K Hayman, Research problems in complex analysis, Bull. Lond. Math. Soc. 16, 490–517 (1984; Zbl 0593.30001)]. Köhler and Tohge’s results are introduced. They also discuss C. C. Yang’s problem on the situation that functions share one value and their derivatives share one value. Chapter 10 introduces research works on the unique range set of meromorphic functions which contain results on meromorphic functions sharing four, three, or two sets. In particular, Gross-Yang’s problem is discussed.

A lot of research papers of uniqueness theory of meromorphic functions appeared recently. Summarizing these results may be a great work, and it may serve as a useful guide to the progress of recent years. I add recent two topics in uniqueness theory of meromorphic functions, one being the studies between this theory and a normal family, see e.g. [M. Fang and L. Zalcman, A note on normality and shared values, J. Aust. Math. Soc. 76, No. 1, 141–150 (2004; Zbl 1074.30032)], another one is the studies with some conditions in angular domains, see, e.g. [J.-H. Zheng, On uniqueness of meromorphic functions with shared values in some angular domains, Can. Math. Bull. 47, No. 1, 152–160 (2004; Zbl 1045.30019)].

Finally, I mention some of most recent results relating with some contents in this book. By Yamanoi, the second fundamental theorem for small functions is obtained, which may give short proofs discussed in Chapter 1, see [K. Yamanoi, The second main theorem for small functions and related problems, Acta Math. 192, No. 1, 225–294 (2004; Zbl 1203.30035)]. In connection with Chapter 9, Hinkkanen’s problem is solved by G. Frank, X. Hua and R. Vaillancourt, see [Meromorphic functions sharing the same zeros and poles, Can. J. Math. 56, No. 6, 1190–1227 (2004; Zbl 1065.30027)].

Reviewer: Katsuya Ishizaki (Saitama)