Taiwanese Journal of Mathematics

ENTIRE FUNCTIONS SHARING ZERO CM WITH THEIR HIGH ORDER DIFFERENCE OPERATORS

Jie Zhang, Jianjun Zhang, and Liangwen Liao

Full-text: Open access

Abstract

In this paper, we investigate uniqueness of entire functions of order less than 2 sharing the value 0 with their difference operators and obtain a result as follows: Let $f$ be a transcendental entire function such that $\sigma{(f)}\lt 2$ and $\lambda(f)\lt \sigma{(f)}$. If $f$ and $\Delta^nf$ share the value $0$ CM, then $f$ must be form of $f(z)=Ae^{\alpha z},$ where $A$ and $\alpha$ are two nonzero constants. This result confirms a conjecture posed earlier on the topic.

Article information

Source
Taiwanese J. Math., Volume 18, Number 3 (2014), 701-709.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706435

Digital Object Identifier
doi:10.11650/tjm.18.2014.3802

Mathematical Reviews number (MathSciNet)
MR3213381

Zentralblatt MATH identifier
1357.30024

Subjects
Primary: 30D35: Distribution of values, Nevanlinna theory 34M10: Oscillation, growth of solutions

Keywords
uniqueness entire function difference equation order

Citation

Zhang, Jie; Zhang, Jianjun; Liao, Liangwen. ENTIRE FUNCTIONS SHARING ZERO CM WITH THEIR HIGH ORDER DIFFERENCE OPERATORS. Taiwanese J. Math. 18 (2014), no. 3, 701--709. doi:10.11650/tjm.18.2014.3802. https://projecteuclid.org/euclid.twjm/1499706435


Export citation

References

  • R. Brück, On entire functions which share one value CM with their first derivative, Results Math., 30 (1996), 21-24.
  • W. Bergweiler and J. K. Langley, Zeros of differences of meromorphic functions, Math. Proc. Cambridge Philos. Soc., 142 (2007), 133-147.
  • Y. M. Chiang and S. J. Feng, On the Nevanlinna characteristic of $f(z+\eta)$ and difference equations in the complex plane, Ramanujian. J., 16 (2008), 105-129.
  • W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
  • W. K. Hayman, Slowly growing integral and subharmonic functions, Comment Math. Helv., 34 (1960), 75-84.
  • I. Laine and C. C. Yang, Value distribution of difference polynomials, Pro Japan Acad Ser A., 83 (2007), 148-151.
  • K. Liu and L. Z. Yang, Value distribution of the difference operator, Arch. Math., 92 (2009), 270-278.
  • C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Science Press, Beijing, Second Printed in 2006.
  • L. Yang, Value Distribution Theory, Springer-Verlag & Science Press, Berlin, 1993.
  • J. Zhang and L. W. Liao, Entire functions sharing a small entire function with their difference operators, Submitted.