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A criterion of normality concerning holomorphic functions whose derivative omit a function II

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Abstract

The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of functions holomorphic on a domain D ⊂ ℂ, all of whose zeros have multiplicity at least k, where k ≥ 2 is an integer. Let h(z) ≢ 0 and ∞ be a meromorphic function on D. Assume that the following two conditions hold for every f ∈ F:

$$ \begin{gathered} (a)f(z) = 0 \Rightarrow |f^{(k)} (z)| < |h(z)|. \hfill \\ (b)f^{(k)} (z) \ne h(z). \hfill \\ \end{gathered} $$

Then F is normal on D.

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References

  1. Clunie, J. and Hayman, W. K., The spherical derivative of integral and meromorphic functions, Comm. Math. Helvet., 40, 1966, 117–148.

    Article  MathSciNet  MATH  Google Scholar 

  2. Liu, X. J. and Ye, Y. S., A criterion of normality concerning holomorphic functions whose derivative omits a function, Chin. Ann. Math., 32B(5), 2011, 699–710.

    Article  MathSciNet  Google Scholar 

  3. Pang, X. C., Bloch’s principle and normal criterion, Sci. China Ser. A, 32, 1989, 782–791.

    MathSciNet  MATH  Google Scholar 

  4. Pang, X. C., Shared values and normal families, Analysis, 22, 2002, 175–182.

    MATH  Google Scholar 

  5. Pang, X. C., Yang, D. G. and Zalcman, L., Normal families of meromorphic functions whose derivative omit a function, Comput. Methods Funct., 2, 2002, 257–265.

    MathSciNet  Google Scholar 

  6. Pang, X. C. and Zalcman, L., Normal families and shared values, Bull. London Math. Soc., 32, 2000, 325–331.

    Article  MathSciNet  MATH  Google Scholar 

  7. Pang, X. C. and Zalcman, L., Normal families of meromorphic functions with multiple zeros and poles, Israel J. Math., 136, 2003, 1–9.

    Article  MathSciNet  MATH  Google Scholar 

  8. Zalcman, L., A heuristic principle in complex function theory, Amer. Math. Monthly, 82, 1975, 813–817.

    Article  MathSciNet  MATH  Google Scholar 

  9. Zalcman, L., Normal families: new perspectives, Bull. Amer. Math. Soc. (N.S.), 35, 1998, 215–230.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, East China Normal University, Shanghai, 200241, China

    Qiaoyu Chen

  2. Department of Mathematics, University of Shanghai for Science and Technology, Shanghai, 200093, China

    Xiaojun Liu

Authors
  1. Qiaoyu Chen
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  2. Xiaojun Liu
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Corresponding author

Correspondence to Qiaoyu Chen.

Additional information

Project supported by the National Natural Science Foundation of China (No. 11071074) and the Outstanding Youth Foundation of Shanghai (No. slg10015).

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Chen, Q., Liu, X. A criterion of normality concerning holomorphic functions whose derivative omit a function II. Chin. Ann. Math. Ser. B 33, 815–822 (2012). https://doi.org/10.1007/s11401-012-0751-y

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  • DOI: https://doi.org/10.1007/s11401-012-0751-y

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