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The Uniqueness of the Entire Functions Whose n-th Powers Share a Small Function with Their Derivatives 被引量：1

The Uniqueness of the Entire Functions Whose n-th Powers Share a Small Function with Their Derivatives
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摘要 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 fn（z） and（fn（z）） share Q（z） CM, then f（z） = ce 1 nz, where c is a nonzero constant. This result extends Lv＇s result from the case of polynomial to small entire function. 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 fn（z） and（fn（z）） share Q（z） CM, then f（z） = ce 1 nz, where c is a nonzero constant. This result extends Lv＇s result from the case of polynomial to small entire function.
作者 Jie ZHANG Hai Yan KANG Liang Wen LIAO
机构地区 College of Science Department of Mathematics
出处 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2014年第5期785-792,共8页 数学学报（英文版）
基金 Supported by the Fundamental Research Funds for the Central Universities(Grant No.2011QNA25) National Natural Science Foundation of China(Grant No.11271179)
关键词 UNIQUENESS entire function share values small function Uniqueness,entire function,share values,small function
分类号 O156 [理学—基础数学]
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参考文献10

  • 1Bruck, R.: On entire functions which share one value CM with their first derivative. Results Math., 30, 21-24 (1996).
  • 2Chang, J. M., Zhu, Y. Z.: Entire functions that share a small function with their derivatives. J. Math. Anal. Appl., 351 491-495 (2009).
  • 3Hayman, W. K.: Meromorphic Functions, Clarendon Press, Oxford, 1964.
  • 4Gundersen, G.: Estimates for the logarithmic derivative of a meromorohic function, plus similar estimates. J. London Math. Soc., Second Set., 27, 88-104 (1988).
  • 5Gundersen, G., Yang, L. Z.: Entire functions that share one value with one or two of their derivatives. J. Math. Anal. Appl., 223, 88-95 (1998).
  • 6Laine, I.: Nevanlinna Theory and Complex Differential Equations, Studies in Math, Vol. 15, de Gruyter, Berlin, 1993.
  • 7Lv, F., Xu, J. F., Chen, A.: Entire functions sharing polynomials with their first derivatives. Arch. Math., 92, 593-601 (2009).
  • 8Yang, C. C., Yi, H. X.: Uniqueness Theory of Meromorphic Functions, Science Press, Beijing, 2006.
  • 9Yang, L.: Value Distribution Theory, Springer-Verlag ~ Science Press, Berlin, 1993.
  • 10Zhang, J. L., Yang, L. Z.: A power of a meromorphic function sharing a small function with its derivative Ann. Acad. Sci. Fenn. Math., 34, 249-260 (2009).

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Acta Mathematica Sinica,English Series
2014年 第5期

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