数学物理学报(英文版) ›› 2012, Vol. 32 ›› Issue (2): 717-723.doi: 10.1016/S0252-9602(12)60051-4

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RESULTS ON A QUESTION OF ZHANG AND YANG

李升|高宗升   

  1. LMIB &|School of Mathematics and Systems Science, Beihang University, Beijing 100191, China;College of Science, Guangdong Ocean University, Zhanjiang 524088, China LMIB &|School of Mathematics and Systems Science, Beihang University, Beijing 100191, China
  • 收稿日期:2010-01-01 修回日期:2011-03-03 出版日期:2012-03-20 发布日期:2012-03-20
  • 基金资助:

    This work is supported by NNSF of China (11171013) and Fundamental Research Funds for the Central Universities NO. 300414. The first author is also supported by the Innovation Foundation of BUAA for Ph.D. Candidates.

RESULTS ON A QUESTION OF ZHANG AND YANG

 LI Sheng, GAO Zong-Sheng   

  1. LMIB &|School of Mathematics and Systems Science, Beihang University, Beijing 100191, China;College of Science, Guangdong Ocean University, Zhanjiang 524088, China LMIB &|School of Mathematics and Systems Science, Beihang University, Beijing 100191, China
  • Received:2010-01-01 Revised:2011-03-03 Online:2012-03-20 Published:2012-03-20
  • Supported by:

    This work is supported by NNSF of China (11171013) and Fundamental Research Funds for the Central Universities NO. 300414. The first author is also supported by the Innovation Foundation of BUAA for Ph.D. Candidates.

摘要:

For a meromorphic function f, let N(l+1(r, 1/f ) denote the counting function of zeros of f of order l at least. Let f be a nonconstant meromorphic function, such that N(r, f) = S(r, f). Denote F = fn. Suppose that F and F′ share 1 CM. If (1) n ≥ 3, or (2) n = 2 and N(r, 1/
f ) = O(N(3(r, 1/f )), then, F = F′, and f assumes the form

f(z) = ce1/n z,
where c is a nonzero constant. This main result of this article gives a positive answer to a question raised by Zhang and Yang [1] for the meromorphic functions case in some sense. And a relative result is proved.

关键词: Nevanlinna theory, shared value, derivative

Abstract:

For a meromorphic function f, let N(l+1(r, 1/f ) denote the counting function of zeros of f of order l at least. Let f be a nonconstant meromorphic function, such that N(r, f) = S(r, f). Denote F = fn. Suppose that F and F′ share 1 CM. If (1) n ≥ 3, or (2) n = 2 and N(r, 1/
f ) = O(N(3(r, 1/f )), then, F = F′, and f assumes the form

f(z) = ce1/n z,
where c is a nonzero constant. This main result of this article gives a positive answer to a question raised by Zhang and Yang [1] for the meromorphic functions case in some sense. And a relative result is proved.

Key words: Nevanlinna theory, shared value, derivative

中图分类号: 

  • 30D35