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Differential inequalities, normality and quasi-normality

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Abstract

We prove that if D is a domain in ℂ, α > 1 and C > 0, then the family F of functions f meromorphic in D such that

$$\frac{{\left| {f'(z)} \right|}} {{1 + \left| {f(z)} \right|^\alpha }} > C for every z \in D$$

is normal in D. For α = 1, the same assumptions imply quasi-normality but not necessarily normality.

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Authors and Affiliations

  1. Department of Mathematics, University of Shanghai for Science and Technology, Shanghai, 200093, P. R. China

    Xiao Jun Liu

  2. Department of Mathematics, Bar-Ilan University, 52900, Ramat-Gan, Israel

    Shahar Nevo

  3. Department of Mathematics, East China Normal University, Shanghai, 200241, P. R. China

    Xue Cheng Pang

Authors
  1. Xiao Jun Liu
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  2. Shahar Nevo
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  3. Xue Cheng Pang
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Corresponding author

Correspondence to Xiao Jun Liu.

Additional information

The first author is supported by National Natural Science Foundation of China (Grant No. 11071074), the Tianyuan Special Funds of the National Natural Science Foundation of China (Grant No. 11226095), Outstanding Youth Foundation of Shanghai (Grant No. slg10015); the second author is supported by the Israel Science Foundation (Grant No. 395/07); the third author is supported by National Natural Science Foundation of China (Grant No. 11071074)

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Liu, X.J., Nevo, S. & Pang, X.C. Differential inequalities, normality and quasi-normality. Acta. Math. Sin.-English Ser. 30, 277–282 (2014). https://doi.org/10.1007/s10114-014-2542-8

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  • DOI: https://doi.org/10.1007/s10114-014-2542-8

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