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On fixed-points and singular values of transcendental meromorphic functions

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Abstract

In this paper we make a further discussion of a relationship between the number of fixed-points and distribution of singular values along the round annuli centered at the origin of a transcendental meromorphic function. To attain our purpose we first establish a fundamental inequality for the modulus of derivative of a holomorphic covering mapping whose image is an annulus by virtue of the hyperbolic metric. The inequality is of independent significance. We make a simple survey on some domain constants for hyperbolic domains.

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References

  1. Beardon A F, Pommerenko C. The Poincaré metric of plane domains. J London Math Soc (2), 1978, 18: 475–483

    Article  MATH  MathSciNet  Google Scholar 

  2. Hayman W K. Meromorphic Functions. Oxford: Clarendon Press, 1964

    MATH  Google Scholar 

  3. Hempel J A. The Poicaré metric on the twice punctured plane and the theroems of Landau and Schottky. J London Math Soc (2), 1979, 20: 435–445

    Article  MATH  MathSciNet  Google Scholar 

  4. Jenkins J A. On explicit bounds in Landau’s theorem II. Canad J Math, 1981, 33: 559–562

    MATH  MathSciNet  Google Scholar 

  5. Kim Y C, Sugawa T. A conformal invariant for nonvanshing analytic functions and its applications. Michigan Math J, 2006, 54: 393–410

    Article  MATH  MathSciNet  Google Scholar 

  6. Langley J K, Zheng J H. On the fixpoints, multiplers and value distribution of certain classes of meromorphic functions. Ann Acad Sci Fenn Math, 1998, 23: 133–150

    MathSciNet  Google Scholar 

  7. Polyá G. On an integral function of an integral function. J London Math Soc, 1926, 1: 12–15

    Article  Google Scholar 

  8. Sugawa T. Poincaré metric and the induced distance of plane domains. In: Proc. of the 10th Internatonal conference on the complex analysis. Korea: Silla University, 2002, 168–175

    Google Scholar 

  9. Sugawa T, Vuorinen M. Some inequalities for the Poincaré metric of plane domains. Math Z, 2005, 250: 885–906

    Article  MATH  MathSciNet  Google Scholar 

  10. Yang L. Value Distribution and New Researches. Berlin: Springer-Verlag, 1993

    Google Scholar 

  11. Zheng J H. Uniformly perfect sets and distribution of holomorphic functions. Nagoya Math J, 2001, 164: 17–33

    MATH  MathSciNet  Google Scholar 

  12. Zheng J H. On multiply-connected Fatou components in iteration of meromorphic functions. J Math Anal Appl, 2006, 313: 24–37

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    JianHua Zheng

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  1. JianHua Zheng
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Correspondence to JianHua Zheng.

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Dedicated to Professor Yang Lo on the Occasion of his 70th Birthday

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Zheng, J. On fixed-points and singular values of transcendental meromorphic functions. Sci. China Math. 53, 887–894 (2010). https://doi.org/10.1007/s11425-010-0036-4

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  • DOI: https://doi.org/10.1007/s11425-010-0036-4

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