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On the value distribution of f2f(k)

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  09 April 2009

Xiaojun Huang  and
Yongxing Gu
Xiaojun Huang
Affiliation:
Mathematics College, Sichuan University, Chengdu, Sichuan 610064, China e-mail: [email protected]
Yongxing Gu
Affiliation:
Department of Mathematics, Chongqing University, Chongqing 400044, China e-mail: [email protected]
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Abstract

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In this paper, we prove that for a transcendental meromorphic function f(z) on the complex plane, the inequality T(r, f) < 6N (r, 1/(f2 f(k)−1)) + S(r, f) holds, where k is a positive integer. Moreover, we prove the following normality criterion: Let ℱ be a family of meromorphic functions on a domain D and let k be a positive integer. If for each ℱ ∈ ℱ, all zeros of ℱ are of multiplicity at least k, and f2 f(k) ≠ 1 for z ∈ D, then ℱ is normal in the domain D. At the same time we also show that the condition on multiple zeros of f in the normality criterion is necessary.

Keywords

meromorphic functionnormal family

MSC classification

Secondary: 30D35: Distribution of values, Nevanlinna theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 78 , Issue 1 , February 2005 , pp. 17 - 26
Copyright
Copyright © Australian Mathematical Society 2005

References

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