Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Zeros and Uniqueness of Difference Polynomials of Meromorphic Functions

  • Received : 2011.10.06
  • Accepted : 2012.02.24
  • Published : 2013.12.23
Download PDF
⟨ Previous Next ⟩

Abstract

This research is a continuation of a recent paper due to the first author in [9]. Different from previous results, we investigate the value distribution of difference polynomials of moromorphic functions in this paper. In particular, we are interested in the existence of zeros of $f(z)^n({\lambda}f(z+c)^m+{\mu}f(z)^m)-a$, where f is a moromorphic function, n, m are two non-negative integers, and ${\lambda}$, ${\mu}$ are non-zero complex numbers. However, the proof here is obviously different to the one in [9]. We also study difference polynomials of entire functions sharing a common value, which improves the result in [10, 13].

Keywords

References

  1. W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana, 11(1995), 355-373.
  2. Y. M. Chiang and S. J. Feng, On the Nevanlinna characteristic of $f(z+{\eta})$ and difference equations in the complex plane, Ramanujan J., 16(1)(2008), 105-129. https://doi.org/10.1007/s11139-007-9101-1
  3. R. G. Halburd and R. J. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn., 31(2006), 463-478.
  4. W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math., 70(1959), 9-42. https://doi.org/10.2307/1969890
  5. J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo and J. L. Zhang, Value sharing results for shifts of meromorphic function, and sufficient conditions for periodicity, J. Math. Anal. Appl., 355(2009), 352-363. https://doi.org/10.1016/j.jmaa.2009.01.053
  6. I. Laine and C. C. Yang, Value Distribution of Difference Polynomials, Proc. Japan Acad. Ser. A Math. Sci., 83(2007), 148-151. https://doi.org/10.3792/pjaa.83.148
  7. K. Liu and L. Z. Yang, Value distribution of the difference operator, Arch. Math. (Basel), 92(2009), 270-278. https://doi.org/10.1007/s00013-009-2895-x
  8. E. Mues, Uber ein Problem von Hayman, Math. Z., 164(1979), 239-259. https://doi.org/10.1007/BF01182271
  9. X. G. Qi and K. Liu, Uniqueness and value distribution of differences of entire functions, J. Math. Anal. Appl., 379(1)(2011), 180-187. https://doi.org/10.1016/j.jmaa.2010.12.031
  10. X. G. Qi, Value distribution and uniqueness of difference polynomials and entire solutions of difference equations, Ann. Polon. Math., 102(2011), 129-142. https://doi.org/10.4064/ap102-2-3
  11. J. F. Xu and H. X. Yi, Uniqueness of entire functions and differential polynomials, Bull. Korean Math. Soc. 44(4)(2007), 623-629. https://doi.org/10.4134/BKMS.2007.44.4.623
  12. C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, 2003.
  13. J. L. Zhang, Value distribution and shared sets of differences of meromorphic functions, J. Math. Anal. Appl., 367(2010), 401-408. https://doi.org/10.1016/j.jmaa.2010.01.038