数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (3): 623-656.doi: 10.1016/S0252-9602(17)30027-9

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NORMAL FAMILY OF MEROMORPHIC FUNCTIONS SHARING HOLOMORPHIC FUNCTIONS AND THE CONVERSE OF THE BLOCH PRINCIPLE

Nguyen Van THIN   

  1. Department of Mathematics, Thai Nguyen University of Education, Luong Ngoc Quyen Street, Thai Nguyen city, Thai Nguyen, Viet Nam
  • 收稿日期:2015-07-23 修回日期:2016-07-01 出版日期:2017-06-25 发布日期:2017-06-25
  • 作者简介:Nguyen Van THIN,E-mail:nguyenvanthintn@gmail.com
  • 基金资助:

    This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2014.41.

NORMAL FAMILY OF MEROMORPHIC FUNCTIONS SHARING HOLOMORPHIC FUNCTIONS AND THE CONVERSE OF THE BLOCH PRINCIPLE

Nguyen Van THIN   

  1. Department of Mathematics, Thai Nguyen University of Education, Luong Ngoc Quyen Street, Thai Nguyen city, Thai Nguyen, Viet Nam
  • Received:2015-07-23 Revised:2016-07-01 Online:2017-06-25 Published:2017-06-25
  • Supported by:

    This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2014.41.

摘要:

In 1996, C. C. Yang and P. C. Hu[8] showed that:Let f be a transcendental meromorphic function on the complex plane, and a ≠0 be a complex number; then assume that n ≥ 2, n1, …, nk are nonnegative integers such that n1 + … + nk ≥ 1; thus
fn(f')n1 … (f(k))nk -a
has infinitely zeros. The aim of this article is to study the value distribution of differential polynomial, which is an extension of the result of Yang and Hu for small function and all zeros of f having multiplicity at least k ≥ 2. Namely, we prove that
fn(f')n1 … (f(k))nk -a(z)
has infinitely zeros, where f is a transcendental meromorphic function on the complex plane whose all zeros have multiplicity at least k ≥ 2, and a(z) ? 0 is a small function of f and n ≥ 2, n1, …, nk are nonnegative integers satisfying n1 + … + nk ≥ 1. Using it, we establish some normality criterias for a family of meromorphic functions under a condition where differential polynomials generated by the members of the family share a holomorphic function with zero points. The results of this article are supplement of some problems studied by J. Yunbo and G. Zongsheng[6], and extension of some problems studied X. Wu and Y. Xu[10]. The main result of this article also leads to a counterexample to the converse of Bloch's principle.

关键词: Normal family, Nevanlinna theory, meromorphic function, sharing function, differential polynomial

Abstract:

In 1996, C. C. Yang and P. C. Hu[8] showed that:Let f be a transcendental meromorphic function on the complex plane, and a ≠0 be a complex number; then assume that n ≥ 2, n1, …, nk are nonnegative integers such that n1 + … + nk ≥ 1; thus
fn(f')n1 … (f(k))nk -a
has infinitely zeros. The aim of this article is to study the value distribution of differential polynomial, which is an extension of the result of Yang and Hu for small function and all zeros of f having multiplicity at least k ≥ 2. Namely, we prove that
fn(f')n1 … (f(k))nk -a(z)
has infinitely zeros, where f is a transcendental meromorphic function on the complex plane whose all zeros have multiplicity at least k ≥ 2, and a(z) ? 0 is a small function of f and n ≥ 2, n1, …, nk are nonnegative integers satisfying n1 + … + nk ≥ 1. Using it, we establish some normality criterias for a family of meromorphic functions under a condition where differential polynomials generated by the members of the family share a holomorphic function with zero points. The results of this article are supplement of some problems studied by J. Yunbo and G. Zongsheng[6], and extension of some problems studied X. Wu and Y. Xu[10]. The main result of this article also leads to a counterexample to the converse of Bloch's principle.

Key words: Normal family, Nevanlinna theory, meromorphic function, sharing function, differential polynomial