On Uniqueness of Two Meromorphic Functions Sharing A Small Function

Abstract

In this paper, we have investigated the uniqueness problems of entire and meromorphic functions concerning differential polynomials sharing a small function. Our results radically extended and improved the results of Bhoosnurmath-Pujari [6] and Harina - Anand [13] not only by sharing small function instead of fixed point but also reducing the lower bound of $ n $. There are some miscalculation in the proof of a result of Harina-Anand [13]. We have corrected all of them in a more convenient way. At last some open questions have been posed for further study in this direction.

Keywords

• meromorphic function,uniqueness theorem,small function,differential polynomials

References

1. [1] M. B. Ahamed, Uniqueness of two differential polynomials of a meromorphic function sharing a set, Commun. Korean Math. Soc., Vol:33, No. 4, (2018), 1181-1203.
2. [2] A. Banerjee and M. B. Ahamed,Results on certain non-linear differential polynomials sharing a small function, Istanbul Univ. Sci. Fac. J. Math. Phys. Astr., Vol: 5, (2014), 17-34.
3. [3] A. Banerjee and M. B. Ahamed,Meromorphic function sharing a small function with its differential polynomials, Acta Univ. Palacki. Olomuc., Fac. rer. nat.,Mathematica, Vol: 54, No. 1, (2015), 33-45.
4. [4] A. Banerjee and M. B. Ahamed,Non-linear differential polynomials sharing a non-zero polynomials with finite weight, Math. Bohemica, Vol: 141, No. 1, (2016), 13-36.
5. [5] A. Banerjee and M. B. Ahamed,Uniqueness of a polynomial and a differential monomial sharing a small function, Analele Univ. de Vest. Timisoara Ser. Math. Info., Vol: 54, No. 1, (2016), 55-71
6. [6] S. S. Bhoosnurmath and V. L. Pujari, Uniqueness of meromorphic functions sharing fixed points, Int. J. Anal., (2013) Art ID: 538027, 12p.
7. [7] M. L. Fang and W. Hong, A unicity theorem for entire functions concerning differential polynomials, Indian J. Pure Appl. Math., Vol:32,(2001), 1343-1348.
8. [8] G. Frank and M. Reinders, A unique range set for meromorphic functions with 11 elements, Complex var. Theory Appl., Vol:37, (1998), 185-193.

9. [9] W. K. Hayman, Meromorphic Functions, Oxford, UK: Clarendon Press, (1964).
10. [10] W. K. Hayman, Research problems in function theory, Athlone Press, London.
11. [11] W. C. Lin and H. X. Yi., Uniqueness theorems for meromorphic functions concerning fixed points, Complex Var. Theory Appl., Vol:49, (2004), 793-806.
12. [12] H. P. Waghamore and N. Shilpa, Generalization of uniqueness theorems for entire and meromorphic functions, Appl. Math., Vol:5, (2014), 1267-1274.
13. [13] H. P. Waghamore and S. Anand, Generalization of uniqueness of meromorphic functions sharing fixed points, Appl. Math., Vol:7, (2016), 939-952.
14. [14] J. F. Xu, F. Lu and H. X. Yi, Fixed points and uniqueness of meromorphic functions, Comput. Math. Appl., Vol:59, (2010), 9-17.
15. [15] H. X. Yi, Meromorphic functions that share two or three values, Kodai Math. J., Vol:13, (1990), 363-372.
16. [16] C. C. Yang and X. Hua, Uniqueness and vale-sharing of meromorphic functions, Ann. Acad. Sci. Fenn. Math., Vol22: (1997), 395-406.
17. [17] C. C. Yang and H. X. Yi, Uniqueness theory of meromorphic functions, Math. Appl., (2003) Kluwer Academic Publishers, Dordrecht.
18. [18] L. Yang, Value distribution theory, Translated and revised from the 1982 Chinese Original, Springer, Berlin.
19. [19] C. C. Yang, On deficiencies of differential polynomials II, Math. Z., Vol:125, (1972), 107-112.