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Milloux theorem, deficiency and fix-points for vector-valued meromorphic functions. (English) Zbl 0919.30022

Many attempts have been made to extend Nevanlinna theory, i.e. the theory of value distribution for functions meromorphic in the complex plane. For example, H. and J. Weyl have initiated the theory of holomorphic and meromorphic curves. The theory of holomorphic curves has later been generalized in order to study equidimensional holomorphic mappings. Nevanlinna theory has also been extended to holomorphic mappings between Riemann surfaces and to certain classes of non holomorphic functions. Around 1970, Ziegler has extended the formalism of Nevanlinna theory to vector-valued meromorphic functions [H. J. W. Ziegler, Vector-valued Nevanlinna theory (1982; Zbl 0496.30026)]. For \(n \geq 1\) and a system of functions \(f_1, \ldots , f_n\) meromorphic in the complex plane one can study the value distribution of \(F=(f_1, \ldots , f_n)\). Combining classical Nevanlinna theory and Ziegler’s results, the author of the present paper proved the Milloux theorem for vector-valued meromorphic functions under general conditions. Next, he applies the former result to establish a theorem on deficiencies and a theorem on the existence of fix-points of meromorphic functions. More precisely, some sufficient conditions are given which assure that the function in question has infinitely many fixed points.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D30 Meromorphic functions of one complex variable (general theory)

Citations:

Zbl 0496.30026
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