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On Sharing Values of Meromorphic Functions and Their Differences

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Abstract

For a meromorphic function f in the complex plane, we prove that if f is a finite order transcendental entire function which has a finite Borel exceptional value a, if \({f(z+\eta)\not\equiv f(z)}\) for some \({\eta\in \mathbb{C}}\), and if f(z + η) − f(z) and f(z) share the value a CM, then

$$ a=0 \quad {\rm and} \quad \frac{f(z+\eta)-f(z)}{f(z)}=A, $$

where A is a nonzero constant. We also consider problems on sharing values of meromorphic functions and their differences when their orders are not an integer or infinite.

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References

  1. Brück R.: On entire functions which share one value CM with their first derivative. Results Math. 30, 21–24 (1996)

    MathSciNet  MATH  Google Scholar 

  2. Chen Z.X., Shon K.H.: On conjecture of R. Brück, concerning the entire function sharing one value CM with its derivative Taiwanese. J. Math. 8(2), 235–244 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Chiang Y.M., Feng S.J.: On the Nevanlinna characteristic of f(z + η) and difference equations in the complex plane. Ramanujan J. 16, 105–129 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gross F.: Factorization of Meromorhpic Functions. U.S. Government Printing Office, Washinton (1972)

    Google Scholar 

  5. Gundersen G.: Meromorphic functions that share four values. Trans. Am. Math. Soc. 277, 545–567 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gundersen G.: Correction to Meromorphic functions that share four values Trans. Am. Math. Soc. 304, 847–850 (1987)

    MathSciNet  MATH  Google Scholar 

  7. Gundersen G., Yang L.Z.: Entire functions that share one values with one or two of their derivatives. J. Math. Anal. Appl. 223(1), 88–95 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hayman W.K.: Meromorphic Functions. Clarendon Press, Oxford (1964)

    MATH  Google Scholar 

  9. Heittokangas J., Korhonen R., Laine I., Rieppo J., Zhang J.: Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity. J Math. Anal. Appl. 355, 352–363 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Heittokangas, J., Korhonen, R., Laine, I., Rieppo, J.: Uniqueness of meromorphic functions sharing values with their shifts. Complex Var. Elliptic Equ. (2011, in press)

  11. Liu K.: Meromorphic functions sharing a set with applications to difference quations. J. Math. Anal. Appl. 359, 384–393 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mues E.: Meromorphic functions sharing four valus. Complex Var. Elliptic Equ. 12, 167–179 (1989)

    MathSciNet  Google Scholar 

  13. Nevanlinna R.: Einige Eindentigkeitssätze in der theorie der meromorphen funktionen. Acta Math. 48, 367–391 (1926)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ozawa M.: On the existence of prime periodic entire functions. Kodai Math. Sem. Rep. 29, 308–321 (1978)

    Article  MATH  Google Scholar 

  15. Yang L.: Value Distribution Theory. Science Press, Beijing (1993)

    MATH  Google Scholar 

  16. Yang C.C., Yi H.X.: Uniqueness Theory of Meromorphic Functions. Kluwer Academic Publishers Group, Dordrecht (2003)

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, People’s Republic of China

    Zong-Xuan Chen

  2. School of Mathematical, Shandong University, Jinan, 250100, People’s Republic of China

    Hong-Xun Yi

Authors
  1. Zong-Xuan Chen
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  2. Hong-Xun Yi
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Corresponding author

Correspondence to Zong-Xuan Chen.

Additional information

This research was supported by the National Natural Science Foundation of China (No: 11171119, 11171184).

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Chen, ZX., Yi, HX. On Sharing Values of Meromorphic Functions and Their Differences. Results. Math. 63, 557–565 (2013). https://doi.org/10.1007/s00025-011-0217-7

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  • DOI: https://doi.org/10.1007/s00025-011-0217-7

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