# SHARED VALUES AND BOREL EXCEPTIONAL VALUES FOR HIGH ORDER DIFFERENCE OPERATORS

• Liao, Liangwen (Department of Mathematics Nanjing University) ;
• Zhang, Jie (College of Science China University of Mining and Technology)
• Published : 2016.01.31
• 85 11

#### Abstract

In this paper, we investigate the high order difference counterpart of $Br{\ddot{u}}ck^{\prime}s$ conjecture, and we prove one result that for a transcendental entire function f of finite order, which has a Borel exceptional function a whose order is less than one, if ${\Delta}^nf$ and f share one small function d other than a CM, then f must be form of $f(z)=a+ce^{{\beta}z}$, where c and ${\beta}$ are two nonzero constants such that $\frac{d-{\Delta}^na}{d-a}=(e^{\beta}-1)^n$. This result extends Chen's result from the case of ${\sigma}(d)$ < 1 to the general case of ${\sigma}(d)$ < ${\sigma}(f)$.

#### Keywords

uniqueness;entire function;difference equation;order

#### Acknowledgement

Supported by : Central Universities, NSF of China

#### References

1. R. Bruck, On entire functions which share one value CM with their first derivative, Results Math. 30 (1996), no. 1-2, 21-24. https://doi.org/10.1007/BF03322176
2. W. Bergweiler and J. K. Langley, Zeros of differences of meromorphic functions, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 1, 133-147. https://doi.org/10.1017/S0305004106009777
3. C. X. Chen and Z. X. Chen, Entire functions and their high order differences, Taiwanese J. Math. 18 (2014), no. 3, 711-729. https://doi.org/10.11650/tjm.18.2014.3453
4. Z. X. Chen and K. H. Shon, On conjecture of R. Bruck, concerning the entire function sharing one value CM with its derivative, Taiwanese J. Math. 8 (2004), no. 2, 235-244. https://doi.org/10.11650/twjm/1500407625
5. 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), no. 1, 105-129. https://doi.org/10.1007/s11139-007-9101-1
6. Y. M. Chiang and S. J. Feng, On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions, Trans. Amer. Math. Soc. 361 (2009), no. 7, 3767-3791. https://doi.org/10.1090/S0002-9947-09-04663-7
7. W. K. Hayman, Slowly growing integral and subharmonic functions, Comment. Math. Helv. 34 (1960), 75-84. https://doi.org/10.1007/BF02565929
8. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
9. 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), no. 1, 352-363. https://doi.org/10.1016/j.jmaa.2009.01.053
10. G. Gundersen, Correction to meromorphic functions that share four values, Trans. Amer. Math. Soc. 304 (1987), no. 2, 847-850.
11. G. Gundersen, Finite order solutions of second order linear differential equations, Trans. Amer. Math. Soc. 305 (1988), no. 1, 415-429. https://doi.org/10.1090/S0002-9947-1988-0920167-5
12. G. Gundersen and L. Z. Yang, Entire functions that share one values with one or two of their derivatives, J. Math. Anal. Appl. 223 (1998), no. 1, 88-95. https://doi.org/10.1006/jmaa.1998.5959
13. E. Mues, Meromorphic functions sharing four valus, Complex Variables Theory Appl. 12 (1989), no. 1-4, 167-179.
14. C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Science Press, Beijing, Second Printed in 2006.
15. L. Yang, Value Distribution Theory, Springer-Verlag & Science Press, Berlin, 1993.
16. J. Zhang, H. Y. Kang, and L. W. Liao, Entire functions sharing a small entire function with their difference operators, Bull. Iran. Math. Soc. accepted.