Advertisement

Computational Methods and Function Theory

, Volume 17, Issue 4, pp 613–634 | Cite as

Uniqueness Theorems for Differential Polynomials Sharing a Small Function

  • Thi Hoai An TaEmail author
  • Viet Phuong Nguyen
Article
  • 236 Downloads

Abstract

Consider meromorphic functions fg,  and \(\alpha ,\) where \(\alpha \) is a small function with respect to f and g. Let Q be a polynomial of one variable. We give suitable conditions on the degree of Q and on the number of zeros and the multiplicities of the zeros of \(Q'\) so as to be able to conclude uniqueness results if differential polynomials of the form \((Q(f))^{(k)}\) and \((Q(g))^{(k)}\) share \(\alpha \) counting multiplicities. We do not assume that Q has a large order zero, nor do we place restrictions on the zeros and poles of \(\alpha .\) Thus, our work improves on many prior results that either assume Q has a high order zero or place restrictions on the small function \(\alpha \).

Keywords

Meromorphic functions Entire functions Nevanlinna theory Uniqueness Sharing value Differential polynomial 

Mathematics Subject Classification

30D35 

Notes

Acknowledgements

We would like to thank the referees for useful suggestions.

References

  1. 1.
    An, T.T.H., Diep, N.T.N.: Genus one factors of curves defined by separated variable polynomials. J. Number Theory 133, 2616–2634 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    An, T.T.H., Wang, J.T.-Y.: Uniqueness polynomials for complex meromorphic functions. Int. J. Math. 13(10), 1095–1115 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    An, T.T.H., Wang, J.T.Y., Wong, P.M.: Strong uniqueness polynomials: the complex case. Complex Var. 49(1), 25–54 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Avanzi, R.M., Zannier, U.M.: Genus one curves defined by separated variable polynomials and a polynomial Pell equation. Acta Arith. 99(3), 227–256 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Avanzi, R.M., Zannier, U.M.: The equation \(f(X)=f(Y)\) in rational functions \(X=X(t)\), \(Y=Y(t)\). Compos. Math. 139(3), 263–295 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bergweiler, W., Eremenko, A.: On the singularities of the inverse to a meromorphic function of finite order. Rev. Mat. Iberoam. 11(2), 355–373 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bhoosnurmatha, S.S., Dyavanal, R.S.: Uniqueness and value-sharing of meromorphic functions. Comput. Math. Appl. 53, 1191–1205 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Boussaf, K., Escassut, A., Ojeda, J.: Complex meromorphic functions \(f ^{\prime }P^{\prime }( f ), g^{\prime }P^{\prime }(g)\) sharing a small function. Indag. Math. 24, 15–41 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cherry, W., Ye, Z.: Nevanlinna’s theory of value distribution. The second main theorem and its error terms. In: Springer Monographs in Mathematics. Springer, Berlin, xii+201 pp, (2001)Google Scholar
  10. 10.
    Chen, H.H., Fang, M.L.: On the value distribution of \(f^nf^{\prime }\). Sci. China Ser. A 38, 789–798 (1995)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Escassut, A.: Meromorphic functions of uniqueness. Bull. Sci. Math. 131(3), 219–241 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fang, M., Hong, W.: A unicity theorem for entire functions concerning differential polynomials. Indian J. Pure Appl. Math. 32(9), 1343–1348 (2001)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Fang, M., Hua, X.H.: Entire functions that share one value. J. Nanjing Univ. Math. Biq. 13(1), 44–48 (1996)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Fujimoto, H.: On uniqueness of meromorphic functions sharing finite sets. Am. J. Math. 122(6), 1175–1203 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fujimoto, H.: On uniqueness polynomials for meromorphic functions. Nagoya Math. J. 170, 33–46 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hayman, W.K.: Picard values of meromorphic functions and their derivatives. Ann. Math. 70, 9–42 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hayman, W.K.: Meromorphic Functions. Oxford Mathematical Monographs Clarendon Press, Oxford (1964)Google Scholar
  18. 18.
    Hayman, W.K.: Research Problems in Function Theory. The Athlone Press, London (1967)zbMATHGoogle Scholar
  19. 19.
    Mues, E.: Über ein Problem von Hayman (German). Math. Z. 164(3), 239–259 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lin, W.C., Yi, H.X.: Uniqueness theorems for meromorphic functions concerning fixed-points. Complex Var. Theory Appl. 49(11), 793–806 (2004)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Ru, M.: Nevanlinna Theory and Its Relation to Diophantine Approximation. World Scientific, Singapore (2001)CrossRefzbMATHGoogle Scholar
  22. 22.
    Xu, J.F., Lü, F., Yi, H.X.: Fixed-points and uniqueness of meromorphic functions. Comput. Math. Appl. 59, 9–17 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Yang, C.C.: On deficiencies of differential polynomials II. Math. Z. 149, 107–112 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Yang, C.C., Hua, X.: Uniqueness and value sharing of meromorphic functions. Ann. Acad. Sci. Fenn. Math. 22, 395–406 (1997)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Zhang, J.L.: Uniqueness theorems for entire functions concerning fixed-points. Comput. Math. Appl. 56, 3079–3087 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhang, X.Y., Chen, J.F., Lin, W.C.: Entire or meromorphic functions sharing one value. Comput. Math. Appl. 56, 1876–1883 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zhang, X.B., Xu, J.F.: Uniqueness of meromorphic functions sharing a small function and its applications. Comput. Math. Appl. 61, 722–730 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam
  2. 2.Institute of Mathematics and Applied Sciences (TIMAS)Thang Long UniversityHanoiVietnam
  3. 3.Thai Nguyen University of Economics and Business AdministrationThai NguyenVietnam

Personalised recommendations