×
Xuan, Zu-Xing; Wu, Nan

On the Nevanlinna’s theory for vector-valued mappings. (English) Zbl 1194.30035

Abstr. Appl. Anal. 2010, Article ID 864539, 15 p. (2010).
Summary: The purpose of this paper is to establish the first and second fundamental theorems for \(E\)-valued meromorphic mappings from a generic domain \(D\subset \mathbb C\) to an infinite dimensional complex Banach space \(E\) with a Schauder basis. This is a continuation of the work of C.-G. Hu and Q. Hu [Complex Var. Elliptic Equ. 51, No. 8–11, 777–791 (2006; Zbl 1183.30027)]. For \(f(z)\) defined in the disk, we will prove Chuang’s inequality, which compares the relationship between \(T(r,f)\) and \(T(r,f^{\prime})\). Consequently, we obtain that the order and the lower order of \(f(z)\) and its derivative \(f^{\prime}(z)\) are the same.

Cited in 2 Documents

MSC:

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

Keywords:

meromorphic vector-valued mapping; order of a mapping; lower order of a mapping

Citations:

Zbl 1183.30027
Cite Review PDF
Full Text: DOI EuDML
Creative Commons CC license icon Creative Commons BY license icon

References:

[1] H. Ziegler, Vector Valued Nevanlinna Theory, vol. 73 of Research Notes in Mathematics, Pitman, Boston, Mass, USA, 1982. · Zbl 0496.30026
[2] I. Lahiri, “Milloux theorem and deficiency of vector-valued meromorphic functions,” The Journal of the Indian Mathematical Society, vol. 55, no. 1-4, pp. 235-250, 1990. · Zbl 0734.30030
[3] I. Lahiri, “Generalisation of an inequality of C. T. Chuang to vector meromorphic functions,” Bulletin of the Australian Mathematical Society, vol. 46, no. 2, pp. 317-333, 1992. · Zbl 0774.30035 · doi:10.1017/S000497270001193X
[4] I. Lahiri, “Milloux theorem, deficiency and fix-points for vector-valued meromorphic functions,” The Journal of the Indian Mathematical Society, vol. 59, no. 1-4, pp. 45-60, 1993. · Zbl 0919.30022
[5] C.-G. Hu and Q. Hu, “The Nevanlinna’s theorems for a class,” Complex Variables and Elliptic Equations, vol. 51, no. 8-11, pp. 777-791, 2006. · Zbl 1183.30027 · doi:10.1080/17476930600667510
[6] W. Arendt and N. Nikolski, “Vector-valued holomorphic functions revisited,” Mathematische Zeitschrift, vol. 234, no. 4, pp. 777-805, 2000. · Zbl 0976.46030 · doi:10.1007/s002090000103
[7] K.-G. Grosse-Erdmann, “The locally convex topology on the space of meromorphic functions,” Australian Mathematical Society A, vol. 59, no. 3, pp. 287-303, 1995. · Zbl 0853.30030 · doi:10.1017/S1446788700037198
[8] E. Jord, “Topologies on spaces of vector-valued meromorphic functions,” Journal of the Australian Mathematical Society, vol. 78, no. 2, pp. 273-290, 2005. · Zbl 1082.46028 · doi:10.1017/S1446788700008089
[9] A. A. Goldberg and I. V. Ostrovskii, Value Distribution of Meromorphic Functions, vol. 236 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 2008. · Zbl 1152.30026
[10] J. Zheng, Value Distribution of Meromorphic Functions, Tsinghua University Press, Beijing, China, 2010. · Zbl 1233.30004
[11] L. Yang, Value Distribution And New Research, Springer, Berlin, Germany, 1993.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.