Taiwanese Journal of Mathematics

Value Distribution of Products of Meromorphic Functions and Their Differences

Zong-Xuan Chen

Full-text: Open access

Abstract

In this paper, we study zeros of difference product $f(z)^n \Delta f(z)$ ($n \geq 2$), and the value distribution of difference product $f(z) \Delta f(z)$, where $f(z)$ is a transcendental entire function of finite order, $\Delta f(z) = f(z+c)-f(z)$, where $c$ ($\not= 0$) is a constant such that $f(z+c) \not\equiv f(z)$.

Article information

Source
Taiwanese J. Math., Volume 15, Number 4 (2011), 1411-1421.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406353

Digital Object Identifier
doi:10.11650/twjm/1500406353

Mathematical Reviews number (MathSciNet)
MR2848883

Zentralblatt MATH identifier
1239.30015

Subjects
Primary: 30D35: Distribution of values, Nevanlinna theory 39A10: Difference equations, additive

Keywords
difference zero entire function

Citation

Chen, Zong-Xuan. Value Distribution of Products of Meromorphic Functions and Their Differences. Taiwanese J. Math. 15 (2011), no. 4, 1411--1421. doi:10.11650/twjm/1500406353. https://projecteuclid.org/euclid.twjm/1500406353


Export citation

References

  • M. Ablowitz, R. G. Halburd and B. Herbst, On the extension of Painlev${\rm \acute{e}}$ property to difference equations, Nonlinearty, 13 (2000), 889-905.
  • W. Bergweiler and J. K. Langley, Zeros of differences of meromorphic functions, Math. Proc. Camb. Phil. Soc., 142 (2007), 133-147.
  • Z. X. Chen and K. H. Shon, On zeros and fixed points of differencers of meromorphic functions, J. Math. Anal. Appl., 344 (2008), 373-383.
  • Z. X. Chen and K. H. Shon, Estimates for zeros of differences of meromorphic functions, Science in China Series A, 52(11) (2009), 2447-2458.
  • Z. X. Chen and K. H. Shon, Value distribution of meromorphic solutions of certain difference Painlev${\rm \acute{e}}$ equations, J. Math. Anal. Appl., 364 (2010), 556-566.
  • 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 (2008), 105-129.
  • J. Clunie, On a result of Hayman, J. London Math. Soc., 42 (1967), 389-392.
  • R. G. Halburd and R. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl., 314 (2006), 477-487.
  • R. G. Halburd and R. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math., 31 (2006), 463-478.
  • W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
  • W. K. Hayman, Picard value of meromorphic functions and and their derivaties, Annals of Math., 70 (1959), 9-42.
  • J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo and K. Tohge, Complex difference equations of Malmquist type, Comput. Methods Funct. Theory, 1 (2001), 27-39.
  • 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.
  • K. Ishizaki and N. Yanagihara, Wiman-Valiron method for difference equations, Nagoya Math. J., 175 (2004), 75-102.
  • I. Laine and Chung-Chun Yang, Value distribution of difference polynomials, Proc. Japan Acad., 83A (2007), 148-151.
  • K. Liu and L. Z. Yang, Value distribution of the difference operator, Arch. Math., 92 (2009), 270-278.
  • L. Yang, Value Distribution Theory, Science Press, Beijing, 1993.
  • C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers Group, Dordrecht, 2003.