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**Value distribution theory.**
*(English)*
Zbl 0790.30018

Berlin: Springer-Verlag. Beijing: Science Press, xii, 269 p. (1993).

This text book of value distribution theory (= Nevanlinna theory = NT) is a thorough revision of an earlier version [Beijing: Science Press (1982; Zbl 0633.30029)]. The book is very carefully and skillfully written. I do not know of any clearer and easier to follow exposition of NT than the 41 pages of Chapter 1. These include the Osgood-Steinmetz generalisation of the second fundamental theorem to “small” functions.

Chapter 2 gives the basic facts of the theory of normal functions following the method of Montel. The Theorems of Landau, Schottky and Picard are proved in it.

Chapter 4 gives an up-to-date account of the relations between the characteristic function \(T(r,f)\) and \(T(r,f')\). It includes a simplified version of the proof of the theorem: A family of functions \({\mathcal F}\) meromorphic in a region \(D\) is normal in \(D\), if for some natural integer \(k\) and all \(F \in {\mathcal F}\), \(f(z) (f^{(k)} (z)-1)=0\) in \(D\). This theorem is due to Gu Yongxing.

Chapter 7 explains the Baernstein \(T^*\)-function and gives some of its applications, like the “Ellipse Theorem”. The remaining chapters 3, 5 and 6 bring a very complete account of results centering on the notions of Borel directions and of Julia lines. Many of the results are due to Yang Lo and Zhang Guanghou, especially those concerning relations between the number of Julia lines and the number of deficient values of meromorphic functions of finite positive order. There are considerable simplifications compared with the original proofs.

Because of its good exposition the book is easy to follow step-by-step. But the proofs are severely analytical. Geometrical notions, like hyperbolic geometry are not used at all, others, like spherical distance on the Riemann sphere, are only mentioned in passing. To gain a wider and more balanced view of NT the reader of the book should also read [Steven G. Krantz, Complex analysis: the geometric viewpoint. Washington: The Mathematical Association of America (1990; Zbl 0743.30002)].

Chapter 2 gives the basic facts of the theory of normal functions following the method of Montel. The Theorems of Landau, Schottky and Picard are proved in it.

Chapter 4 gives an up-to-date account of the relations between the characteristic function \(T(r,f)\) and \(T(r,f')\). It includes a simplified version of the proof of the theorem: A family of functions \({\mathcal F}\) meromorphic in a region \(D\) is normal in \(D\), if for some natural integer \(k\) and all \(F \in {\mathcal F}\), \(f(z) (f^{(k)} (z)-1)=0\) in \(D\). This theorem is due to Gu Yongxing.

Chapter 7 explains the Baernstein \(T^*\)-function and gives some of its applications, like the “Ellipse Theorem”. The remaining chapters 3, 5 and 6 bring a very complete account of results centering on the notions of Borel directions and of Julia lines. Many of the results are due to Yang Lo and Zhang Guanghou, especially those concerning relations between the number of Julia lines and the number of deficient values of meromorphic functions of finite positive order. There are considerable simplifications compared with the original proofs.

Because of its good exposition the book is easy to follow step-by-step. But the proofs are severely analytical. Geometrical notions, like hyperbolic geometry are not used at all, others, like spherical distance on the Riemann sphere, are only mentioned in passing. To gain a wider and more balanced view of NT the reader of the book should also read [Steven G. Krantz, Complex analysis: the geometric viewpoint. Washington: The Mathematical Association of America (1990; Zbl 0743.30002)].

Reviewer: W. H. J. Fuchs (Ithaca)