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Normal Families and Uniqueness of Entire Functions and Their Derivatives

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Abstract

Let f be a nonconstant entire function; let k ≥ 2 be a positive integer; and let a be a nonzero complex number. If f(z) = a ⇒. f'(z) = a, and f'(z) = a ⇒. f (k)(z) = a, then either f = Ce λz + a or f = Ce λz + a(λ - 1)/λ, where C and λ are nonzero constants with λ k-1 = 1. The proof is based on the Wiman–Valiron theory and the theory of normal families in an essential way.

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Authors and Affiliations

  1. Department of Mathematics, Changshu Institute of Technology, Changshu 215500, P. R. China

    Jiang Ming Chang

  2. Department of Mathematics, Nanjing Normal University, Nanjing 210097, P. R. China

    Jiang Ming Chang

  3. Department of Applied Mathematics, South China Agricultural University, Guangzhou 510642, P. R. China

    Ming Liang Fang

  4. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100800, P. R. China

    Ming Liang Fang

Authors
  1. Jiang Ming Chang
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  2. Ming Liang Fang
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Correspondence to Jiang Ming Chang.

Additional information

Supported by the NNSF of China (Grant No. 10471065), the NSF of Education Department of Jiangsu Province (Grant No. 04KJD110001), the SRF for ROCS, SEM., and the Presidential Foundation of South China Agricultural University

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Chang, J.M., Fang, M.L. Normal Families and Uniqueness of Entire Functions and Their Derivatives. Acta Math Sinica 23, 973–982 (2007). https://doi.org/10.1007/s10114-005-0861-5

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  • DOI: https://doi.org/10.1007/s10114-005-0861-5

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