Archiv der Mathematik

, Volume 97, Issue 5, pp 475–483 | Cite as

Entire functions sharing one or two finite values CM with their shifts or difference operators

Article

Abstract

We prove some results on the uniqueness of entire functions sharing one or two finite values CM with their shifts or difference operators. Our results include shifted and difference analogues of the Brück conjecture.

Mathematics Subject Classification (2010)

Primary 30D35 Secondary 39A05 

Keywords

Brück conjecture Shared value Derivative Shift Difference operator 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brück R.: On entire functions which share one value CM with their first derivative. Results Math. 30, 21–24 (1996)MathSciNetMATHGoogle Scholar
  2. 2.
    Chen Z.X.: The growth of solutions of f′′ + e z f′ + Q(z)f = 0 where the order of Q = 1. Science in China (Ser.A) 45, 290–300 (2002)MATHGoogle Scholar
  3. 3.
    Chen Z.X., Shon K.H.: On conjecture of R. Brück concernig the entire function sharing one value CM with its derivative. Taiwanese J. Math. 8, 235–244 (2004)MathSciNetMATHGoogle Scholar
  4. 4.
    Chiang Y.M., Feng S.J.: On the Nevanlinna characteristic f(z + η) and difference equations in the complex plane. Ramanujan J. 16, 105–129 (2008)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chiang Y.M., Feng S.J.: On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. Trans. Amer. Math. Soc. 361, 3767–3791 (2009)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    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)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Halburd R.G., Korhonen R.J.: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn. Math. 31, 463–478 (2006)MathSciNetMATHGoogle Scholar
  8. 8.
    Hayman W.K.: Meromorphic Functions. Clarendon Press, Oxford (1964)MATHGoogle Scholar
  9. 9.
    G. G. Gundersen, Meromorphic functions that share finite values with their derivative. J. Math. Anal. Appl. 75 (1980), 441–446. (Correction: 86 (1982), 307).Google Scholar
  10. 10.
    Gundersen G.G.: Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. J. London Math. Soc. 37, 88–104 (1988)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Gundersen G.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)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Heittokangas J. et al.: Uniqueness of meromorphic functions sharing values with their shifts. Complex Var. Elliptic Equ. 56, 81–92 (2011)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Heittokangas J. et al.: Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity. J. Math. Anal. Appl. 355, 352–363 (2009)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Laine I.: Nevanlinna Theory and Complex Differential Equations. Walter de Gruyter, Berlin (1993)CrossRefGoogle Scholar
  15. 15.
    Li S., Gao Z.S.: A note on the Brück Conjecture. Arch. Math. 95, 257–268 (2010)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Liu K., Yang L.Z.: Value distribution of the difference operator. Arch. Math. 92, 270–278 (2009)MATHCrossRefGoogle Scholar
  17. 17.
    Mues E., Steinmetz N.: Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen. Manuscripta Math. 29, 195–206 (1979)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Ozawa M.: On the existence of prime periodic entire functions. Kodai Math. Sem. Rep. 29, 308–321 (1978)MATHCrossRefGoogle Scholar
  19. 19.
    Rubel L.A., Yang C.C.: Values shared by an entire function and its derivative. Lecture Notes in Math. 599, 101–103. Springer-Verlag, Berlin (1977)Google Scholar
  20. 20.
    Yang C.C., Yi H.X.: Uniqueness theory of meromorphic functions. Kluwer Academic publishers, The Netherlands (2003)MATHGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.LMIB and School of Mathematics and Systems ScienceBeihang UniversityBeijingPeople’s Republic of China
  2. 2.College of ScienceGuangdong Ocean UniversityZhanjiangPeople’s Republic of China

Personalised recommendations