Uniqueness of Meromorphic Functions Sharing Values with Their nth Order Exact Differences

  • Z. Gao
  • R. KorhonenEmail author
  • J. Zhang
  • Y. Zhang


Let f(z) be a transcendental meromorphic function in the complex plane of hyper-order strictly less than 1. It is shown that if f(z) and its nth exact difference Δnf(z) (≢ 0) share three distinct periodic functions \({\rm{a, b, c}} \in \mathcal{\hat{S}}(f)\) with period 1 CM, where \(\mathcal{\hat{S}}(f) = \mathcal{S}(f)\cup\{{\infty}\}\) and \(\mathcal{S}(f)\) denotes the set of all small functions of f(z), then Δnf(z) ≡ f(z).

Key words and phrases

uniqueness meromorphic function exact difference sharing values 

Mathematics Subject Classification

30D35 39A10 


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  1. [1]
    Z. X. Chen and H. X. Yi, On sharing values of meromorphic functions and their differences, Results Math., 63 (2013), 557–565.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Y. M. Chiang and S. J. Feng, On the Nevanlinna characteristic of f(z + η) and difference equations in the complex plane, Ramanujan J., 16 (2008), 105–129.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    G. Frank and G. Weissenborn, Meromorphe Funktionen, die mit einer ihrer Ableitungen Werte teilen, Complex Variables Theory Appl., 7 (1986), 33–43.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    G. G. Gundersen, Meromorphic functions that share two finite values with their derivative, Pacific J. Math., 105 (1983), 299–309.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    R. G. Halburd and R. J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl., 314 (2006), 477–487.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    R. G. Halburd and R. J. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math., 31 (2006), 463–478.MathSciNetzbMATHGoogle Scholar
  7. [7]
    R. Halburd, R. Korhonen and K. Tohge, Holomorphic curves with shift-invariant hyperplane preimages, Trans. Amer. Math. Soc., 366 (2014), 4267–4298.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    J. Heittokangas, R. Korhonen, I. Laine and J. Rieppo, Uniqueness of meromorphic functions sharing values with their shifts, Complex Var. Elliptic Equ., 56 (2011), 81–92.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo and J. Zhang, Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity, J. Math. Anal. Appl., 355 (2009), 352–363.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    I. Laine, Nevanlinna Theory and Complex Differential Equations, de Gruyter Studies in Mathematics, Vol. 15, Walter de Gruyter & Co. (Berlin, 1993).CrossRefGoogle Scholar
  11. [11]
    S. Lang, Introduction to Complex Hyperbolic Spaces, Springer-Verlag (New York, 1987).CrossRefzbMATHGoogle Scholar
  12. [12]
    P. Li, Unicity of meromorphic functions and their derivatives, J. Math. Anal. Appl., 285 (2003), 651–665.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    S. Li and Z. S. Gao, A note on the Brück conjecture, Arch. Math. (Basel), 95 (2010), 257–268.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    S. Li and Z. S. Gao, Entire functions sharing one or two finite values CM with their shifts or difference operators, Arch. Math. (Basel), 97 (2011), 475–483.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    X. M. Li, H. X. Yi and C. Y. Kang, Results on meromorphic functions sharing three values with their difference operators, Bull. Korean Math. Soc., 52 (2015), 1401–1422.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    P. Li and W. J. Wang, Entire functions that share a small function with its derivative, J. Math. Anal. Appl., 328 (2007), 743–751.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    A. Z. Mohonko, The Nevanlinna characteristics of certain meromorphic functions, Teor. Funkciĭ Funkcional. Anal. i Prilŏzen., (14) (1971), 83–87 (in Russian).MathSciNetGoogle Scholar
  18. [18]
    E. Mues and N. Steinmetz, Meromorphe Funktionen, die mit ihrer Ableitung zwei Werte teilen, Results Math., 6 (1983), 48–55.zbMATHGoogle Scholar
  19. [19]
    L. A. Rubel and C. C. Yang, Values shared by an entire function and its derivative, in: Complex Analysis (Proc. Conf., Univ. Kentucky, Lexington, KY, 1976), Lecture Notes in Math., Vol. 599, Springer (Berlin, 1977), pp. 101–103.Google Scholar
  20. [20]
    C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Mathematics and its Applications, Vol. 557, Kluwer Academic Publishers Group (Dordrecht, 2003).CrossRefGoogle Scholar
  21. [21]
    G. Valiron, Sur la dérivée des fonctions algébröıdes, Bull. Soc. Math. France, 59 (1931), 17–39.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    J. Zhang and L. W. Liao, Entire functions sharing some values with their difference operators, Sci. China Math., 57 (2014), 2143–2152.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.School of Mathematics and Systems ScienceBeihang University, LMIBBeijingP. R. China
  2. 2.Department of Physics and MathematicsUniversity of Eastern FinlandJoensuuFinland

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