Skip to main content
SpringerLink
Log in

Zeros and uniqueness of Q-difference polynomials of meromorphic functions with zero order

  • Published:
Proceedings - Mathematical Sciences Aims and scope Submit manuscript

Abstract

In this paper, we investigate the value distribution of q-difference polynomials of meromorphic function of finite logarithmic order, and study the zero distribution of difference-differential polynomials [f n(z)f(qz + c)](k) and [f n(z)(f(qz + c)−f(z))](k), where f(z) is a transcendental function of zero order. The uniqueness problem of difference-differential polynomials is also considered.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barnett D C, Halburd R G, Korhonen R J and Morgan W, Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations, Proc. Roy. Soc. Edinburgh A137 (2007) 457–474

  2. Chern P T Y, On meromorphic functions with finite logarithmic order, Trans. Amer. Math. Soc. 358(2) (2006) 473–489

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

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

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

  6. Hayman W K, Picard values of meromorphic functions and their derivatives, Ann. Math. 70 (1959) 9–42

  7. Lahiri I and Sarkar A, Uniqueness of meromorphic function and its derivative, J. Inequal. Pure. Appl. Math. 5(1) (2004) 1–20

  8. Liu K and Qi X G, Meromorphic solutions of q-difference equations, Ann. Pol. Math. 101(3) (2011) 215–225

  9. Liu K, Liu X L and Cao T B, Some results on zeros and uniqueness of difference-differential polynomials, Appl. Math. J. Chinese Univ. 27 (2012) 94–104

  10. Liu K, Liu X L and Cao T B, Uniqueness and zeros of q-shift difference polynomials, Proc. Indian Acad. Sci. (Math. Sci.) 121(3) (2011) 301–310

  11. Wang Y F and Fang M L, Picard values and normal families of meromorphic functions with multiple zeros, Acta Math. Sinica. 14(1) (1998) 17–26

  12. Xu J F and Zhang X B, The zeros of q-difference polynomials of meromorphic functions, Advances in Difference Equations. (2012) 2012:200 1-10

  13. Yang C C and Hua X H, Uniqueness and value sharing of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 22 (1997) 395–406

  14. Yi H X and Yang C C, Uniqueness theory of meromorphic functions (2003) (Kluwer: Science Press 1995)

  15. Zhang J L and Korhonen R J, On the Nevanlinna characteristic of f(qz) and its applications, J. Math. Anal. Appl. 369 (2010) 537–544

Download references

Acknowledgements

The authors would like to thank the anonymous referee for valuable suggestions and comments including the Remark in §4 to improve the paper. This research was partly supported by the NSFC (11101201, 11301260), the NSF of Jiangxi (20122BAB211001, 2013BAB211003), Foundation of Post Ph.D. of Jiangxi (2013KY10), and the NSF of the Education Department of Jiangxi (GJJ13077, GJJ13078) of People’s Republic of China.

Author information

Authors and Affiliations

  1. Department of Mathematics, Nanchang University, Nanchang, Jiangxi, 330031, People’s Republic of China

    TING-BIN CAO, KAI LIU & NA XU

Authors
  1. TING-BIN CAO
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. KAI LIU
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. NA XU
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to TING-BIN CAO.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

CAO, TB., LIU, K. & XU, N. Zeros and uniqueness of Q-difference polynomials of meromorphic functions with zero order. Proc Math Sci 124, 533–549 (2014). https://doi.org/10.1007/s12044-014-0196-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12044-014-0196-1

Keywords

2010 Mathematics Subject Classification.

Search

Navigation