Value Distribution and Uniqueness of Entire Functions Related to Difference Polynomial

Year 2016, Volume: 4 Issue: 2, 29 - 36, 30.10.2016

Pulak Sahoo , Sajahan Seikh

https://doi.org/10.36753/mathenot.421449

Abstract

In this paper, we investigate the distribution of zeros of certain type of difference polynomial. At the
same time we also investigate the uniqueness results when two difference products of entire functions
share one value counting or ignoring multiplicities by considering that the functions share the value zero,
counting multiplicities. The results of the paper improve and generalize some recent concerning results
of W.L. Li and X.M. Li [Bull. Malay. Math. Sci. Soc., 39(2016), 499-515]. 

Keywords

Uniqueness, entire function, difference polynomial

References

• [1] A. Banerjee, Meromorphic functions sharing one value. Int. J. Math. Math. Sci. 22 (2005), 3587-3598.
• [2] W. Bergweiler and J.K. Langley, Zeros of differences of meromorphic functions. Math. Proc. Camb. Phil. Soc. 142 (2007), 133-147.
• [3] Y.M. Chiang and S.J. Feng, On the Nevanlinna characteristic of f(z +η) and difference equations in the complex plane. Ramanujan J. 16 (2008), 105-129.
• [4] M.L. Fang and X.H. Hua, Entire functions that share one value. J. Nanjing Univ. Math. Biquarterly 13 (1996), 44-48.
• [5] R.G. Halburd and R.J. Korhonen, Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn. Math. 31 (2006), 463-478.
• [6] R.G. Halburd and R.J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with application to difference equations. J. Math. Anal. Appl. 314(2006), 477-487.
• [7] W.K. Hayman, Meromorphic Functions. The Clarendon Press, Oxford (1964).
• [8] I. Laine, Nevanlinna Theory and Complex Differential Equations. Walter de Gruyter, Berlin/Newyork, 1993.
• [9] I. Laine and C.C. Yang, Value distribution of difference polynomials. Proc. Japan Acad. SerA Math. Sci. 83 (2007), 148-151.
• [10] W.L. Li and X.M. Li, Results on uniqueness of entire functions related to difference polynomial. Bull. Malay. Math. Sci. Soc. 39 (2016), 499-515.
• [11] X. Luo and W.C. Lin, Value sharing results for shifts of meromorphic functions. J. Math. Anal. Appl. 377 (2011), 441-449.
• [12] X.G. Qi, L.Z. Yang and K. Liu, Uniqueness and periodicity of meromorphic functions concerning the difference operator. Comput. Math. Appl. 60 (2010), 1739-1746.
• [13] P. Sahoo, Entire functions that share fixed points with finite weights. Bull. Belgian Math. Soc. - Simon Stevin 18 (2011), 883-895.
• [14] P. Sahoo, Uniqueness and weighted sharing of entire functions. Kyungpook Math. J. 51 (2011), 145-164.
• [15] L. Xudan and W.C. Lin, Value sharing results for shifts of meromorphic functions. J. Math. Anal. Appl. 377 (2011), 441-449.
• [16] C.C. Yang and X.H. Hua, Uniqueness and value sharing of meromorphic functions. Ann. Acad. Sci. Fenn. Math. 22 (1997), 395-406.
• [17] H.X. Yi and C.C. Yang, Uniqueness Theory of Meromorphic Functions, Science Press, Beijing, 1995.
• [18] J.L. Zhang and L.Z. Yang, Some results related to a conjecture of R. Bru¨ck. J. Inequal. Pure Appl. Math. 8 (2007), Art. 18.