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The uniqueness of the entire functions whose n-th powers share a small function with their derivatives

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Abstract

In this paper, we prove the following result: Let f(z) be a transcendental entire function, Q(z) ≢ 0 be a small function of f(z), and n ≥ 2 be a positive integer. If f n(z) and (f n(z))′ share Q(z) CM, then \(f(z) = ce^{\tfrac{1} {n}z}\), where c is a nonzero constant. This result extends Lv’s result from the case of polynomial to small entire function.

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Authors and Affiliations

  1. College of Science, China University of Mining and Technology, Xuzhou, 221116, P. R. China

    Jie Zhang & Hai Yan Kang

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, P. R. China

    Liang Wen Liao

Authors
  1. Jie Zhang
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  2. Hai Yan Kang
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  3. Liang Wen Liao
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Corresponding author

Correspondence to Hai Yan Kang.

Additional information

Supported by the Fundamental Research Funds for the Central Universities (Grant No. 2011QNA25) and National Natural Science Foundation of China (Grant No. 11271179)

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Zhang, J., Kang, H.Y. & Liao, L.W. The uniqueness of the entire functions whose n-th powers share a small function with their derivatives. Acta. Math. Sin.-English Ser. 30, 785–792 (2014). https://doi.org/10.1007/s10114-014-3365-3

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  • DOI: https://doi.org/10.1007/s10114-014-3365-3

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