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**Unicity of meromorphic mappings.**
*(English)*
Zbl 1074.30002

Advances in Complex Analysis and its Applications 1. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1219-5/hbk). ix, 467 p. (2003).

In the book it is treated special topic dealing with meromorphic mappings. This is the unicity theory of meromorphic mappings. There are considered cases of meromorphic functions of one complex variable (chapter 2), meromorphic functions of many complex variables (chapter 3), meromorphic mappings of complex manifolds (chapter 4). In the first chapter the authors introduce necessary results from the Nevanlinna theory that is the basic tool in the book. The book is an advanced course. Perequisites for reading chapters 2-4 are the fundamentals of the Nevanlinna theory for corresponding classes of meromorphic mappings. In the last fifth chapter it is stated the Nevanlinna theory of algebroid functions of several complex variables and its applications. The main notations of the unicity theory of meromorphic functions of one complex variable are the following:

1) functions \(f\) and \(g\) share a CM (counting multiplicity),

2) functions \(f\) and \(g\) share a IM (ignoring multiplicity).

The first means that if \(z_0\) is the zero of multiplicity \(m\) of the equation \(f(z)-a=0\) then \(z_0\) is the same of the equation \(g(z)-a=0.\) Second means that if \(z_0\) is the zero of the equation \(f(z)-a=0\) then \(z_0\) is the same of the equation \(g(z)-a=0.\) The functions \(e^z\) and \(e^{-z}\) share \(0, 1, -1, \infty\) CM. In 1926 R.Nevanlinna proved the following.

1) If two nonconstant meromorphic functions \(f\) and \(g\) share five values \(IM\) then \(f(z)\equiv g(z).\)

2) If two nonconstant meromorphic functions \(f\) and \(g\) share four values CM then either \(f(z)\equiv g(z)\) or \(f\) is a MĂ¶bius transformation of \(g.\)

The unicity theory of meromorphic mappings arose from these two theorems due to many mathematicians especially Chinese ones. One can find the history and results of this theory up to 2003 in the book. Some results of the book are new.

1) functions \(f\) and \(g\) share a CM (counting multiplicity),

2) functions \(f\) and \(g\) share a IM (ignoring multiplicity).

The first means that if \(z_0\) is the zero of multiplicity \(m\) of the equation \(f(z)-a=0\) then \(z_0\) is the same of the equation \(g(z)-a=0.\) Second means that if \(z_0\) is the zero of the equation \(f(z)-a=0\) then \(z_0\) is the same of the equation \(g(z)-a=0.\) The functions \(e^z\) and \(e^{-z}\) share \(0, 1, -1, \infty\) CM. In 1926 R.Nevanlinna proved the following.

1) If two nonconstant meromorphic functions \(f\) and \(g\) share five values \(IM\) then \(f(z)\equiv g(z).\)

2) If two nonconstant meromorphic functions \(f\) and \(g\) share four values CM then either \(f(z)\equiv g(z)\) or \(f\) is a MĂ¶bius transformation of \(g.\)

The unicity theory of meromorphic mappings arose from these two theorems due to many mathematicians especially Chinese ones. One can find the history and results of this theory up to 2003 in the book. Some results of the book are new.

Reviewer: Anatoly Filip Grishin (Khar’kov)

### MSC:

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

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

32H30 | Value distribution theory in higher dimensions |

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

32A20 | Meromorphic functions of several complex variables |

32H04 | Meromorphic mappings in several complex variables |