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Notes on entire functions sharing an entire function of a smaller order with their difference operators

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Abstract

In this paper, we study a uniqueness question of entire functions sharing an entire function of smaller order with their difference operators. The results in this paper extend Theorem 1.1 in [19] by Liu and Yang and deal with Question 1 in [19], where the entire functions are of finite order. Moreover, we repair certain statements in [21] by Li et al., which in turn had depended on questionable assertions of Lemma 2.6 in [20]. Examples are provided to show that the results in this paper are best possible.

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References

  1. Al-Khaladi A.: On meromorphic functions that share one value with their derivative. Analysis 25, 131–140 (2005)

    MathSciNet  MATH  Google Scholar 

  2. Anderson J.M., Clunie J.: Slowly growing meromorphic functions. Comment. Math. Helv. 40, 267–280 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ash R.: Complex variables. Academic Press, New York and London (1971)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  5. Bergweiler W., Langley J.K.: Zeros of differences of meromorphic functions. Math. Proc. Cambridge Philos. Soc. 142, 133–147 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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 

  7. Halburd R.G., Korhonen R.J.: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn. Math. 31, 463–478 (2006)

    MathSciNet  MATH  Google Scholar 

  8. Halburd R.G., Korhonen R.J.: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal, Appl. 314, 477–487 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  10. Hayman W.K.: Slowly growing integral and subharmonic functions. Comment. Math. Helv. 34, 75–84 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  11. Heittokangas J. et al.: Uniqueness of meromorphic functions sharing values with their shifts. Complex Var. Elliptic Equ. 56, 81–92 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Heittokangas J., Korhonen R., Laine I., Rieppo J., Zhang J.L.: 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 

  13. Ishizaki K., Tohge K.: On the complex oscillation of some linear differential equations. J. Math. Anal. Appl. 206, 503–517 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. G. Jank and L.Volkmann, Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen, Birkhäuser, Basel, Boston, 1985

  15. I. Laine Nevanlinna Theory and Complex Differential Equations Walter de Gruyter, Berlin/New York, 1993.

  16. Laine I., Yang C.C.: Clunie theorems for difference and q-difference polynomials. J. London Math. Soc. 76, 556–566 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Laine I., Yang C. C.: Value distribution of difference polynomials. Proc. Japan Acad. Ser. A 83, 148–151 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu K., Yang L.Z.: Value distribution of the difference operator. Arch. Math. 92, 270–278 (2009)

    Article  MATH  Google Scholar 

  20. Li S., Gao Z. S.: A note on the Brück conjecture. Arch. Math. 95, 257–268 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li X.M., Kang C.Y., Yi H.X.: Uniqueness theorems of entire functions sharing a nonzero complex number with their difference operators. Arch. Math. 96, 577–587 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. L. Rubel and C. C. Yang, Values shared by an entire function and its derivative , in “Complex Analysis, Kentucky 1976” (Proc.Conf.), Lecture Notes in Mathematics, Vol 599, pp. 101-103, Springer-Verlag, Berlin, 1977.

  23. J. M. Whittaker, Interpolatory function theory, Cambridge Tract No. 33, Cambridge University Press, Cambridge, 1935.

  24. C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, Dordrecht/Boston/London 2003.

  25. Zhang J.L.: Value distribution and shared sets of differences of meromorphic functions. J. Math. Anal. Appl. 367, 401–408 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Ocean University of China, Qingdao, Shandong, 266100, People’s Republic of China

    Xiao-Min Li & Cong-Yun Kang

  2. Department of Mathematics, Shandong University, Jinan, Shandong, 250100, People’s Republic of China

    Hong-Xun Yi

Authors
  1. Xiao-Min Li
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  2. Hong-Xun Yi
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  3. Cong-Yun Kang
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Corresponding author

Correspondence to Xiao-Min Li.

Additional information

This work is supported by the NSFC (No.11171184), the NSFC & RFBR (Joint Project)(No. 10911120056), the NSF of Shandong Province, China (No. Z2008A01), and the NSF of Shandong Province, China (No. ZR2009AM008).

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Li, XM., Yi, HX. & Kang, CY. Notes on entire functions sharing an entire function of a smaller order with their difference operators. Arch. Math. 99, 261–270 (2012). https://doi.org/10.1007/s00013-012-0425-8

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  • DOI: https://doi.org/10.1007/s00013-012-0425-8

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