Some results on uniqueness of meromorphic functions sharing sets

A. Sarkar

Let f and g be two non constant meromorphic functions defined in the open complex plane C. In 2011 A. Banerjee ([4]), in an attempt to answer on the question of F. Gross ([9]), by W. C. Lin and H. X. Yi ([18]), improved a result of I. Lahiri ([14]) by reducing the cardinality of the set shared by f and g from 7 to 6 under weaker condition on ramification index. In this paper we show that the cardinality of the shared set can further be reduced to 4 as well as the condition on ramification index can be replaced by weaker one to obtain the same conclusion as A. Banerjee ([4]).

1. A. Banerjee, Uniqueness of meromorphic functions that share two sets, Southeast Asian Bull. Math., 31 (2007), 7–17.

2. A. Banerjee, On uniqueness of meromorphic functions that share two sets, Georgian Math. J., 15 (2008), №1, 21–28.

3. A. Banerjee, S. Mukherjee, Uniqueness of meromorphic functions sharing two or three sets, Hokkaido Math. J., 37 (2008), №3, 507–530.

4. A. Banerjee, A uniqueness result on meromorphic functions sharing two sets, J. Adv. Math. Stud., 4 (2011), №1, 1–10.

5. M. Fang, H. Guo, On meromorphic functions sharing two values, Analysis, 17 (1997), №4, 355–366.

6. M. Fang, I. Lahiri, Uniquene range set for certain meromorphic functions, Indian J. Math., 45 (2003), №2, 141–150.

7. M. Fang, W. Xu, A note on a problem of Gross, Chinese Ann. Math. Ser. A, 18 (1997), №4, 355–366.

8. G. Frank, M.R. Reinders, A unique range set for meromorphic functions with 11 elements, Complex Var. Theory Appl., 37 (1998), 185–193.

9. F. Gross, Factorization of meromorphic functions and some open problems, Proc. Conf. Univ. Kentucky, Lexington, Lecture notes in Math., Springer, Berlin, 599 (1977), 51–69.

10. Q. Han, H.X. Yi, Some further results on meromorphic functions that share two sets, Ann. Polon. Math. 91 (2008), №1, 17–31.

11. W.K. Hayman, Meromorphic functions. – Clarendon Press, Oxford, 1964.

12. I. Lahiri, Weighted sharing and uniqueness of meromorphic functions, Nagoya Math. J., 161 (2001), 193–206.

13. I. Lahiri, Weighted value sharing and uniqueness of meromorphic functions, Complex Var. Theory Appl., 46 (2001), 241–253.

14. I. Lahiri, On a question of Hong Xun Yi, Arch Math. (Brno), 38 (2002), 119–128.

15. I. Lahiri, S. Dewan, Value distribution of the product of a meromorphic function and its derivative, Kodai Math. J., 26 (2003), 95–100.

16. P. Li, C.C. Yang, On the unique range sets for meromorphic functions, Proc. Amer. Math. Soc., 124 (1996), 177–185.

17. P. Li, C.C. Yang, Some further results on the unique range sets for meromorphic functions, Kodai Math. J., 18 (1995), 437–450.

18. W.C. Lin, H.X. Yi, Some further results on meromorphic functions that share two sets, Kyungpook Math. J., 43 (2003), 73–85.

19. A.Z. Mohon’ko, On the Nevanlinna characteristics of some meromorphic functions, Theory of Funct. Funct. Anal. Appl., 14 (1971), 83–87.

20. H.X. Yi, Uniqueness of meromorphic functions and a question of Gross, Science in China(A), 37 (1994), №7, 802–813.

21. H.X. Yi, Unicity theorem for meromorphic functions or entire functions, Bull. Austral. Math. Soc. II, 52 (1995), 215–224.

22. H.X. Yi, On a question of Gross concerning uniqueness of entire functions, Bull. Austral. Math. Soc., 57 (1998), 343–349.

23. H.X. Yi, Meromorphic functions that share two sets (Chinese), Acta Math. Sinica (Chinese Ser.), 45 (2002), №1, 75–82.

24. H.X. Yi, Uniqueness of meromorphic functions II, Indian J. Pure Appl. Math., 28 (1997), 509–519.

25. H.X. Yi, W.C. Lin, Uniqueness of meromorphic and a question of Gross, Kyungpook Math. J., 46 (2006), 437–444.

26. H.X. Yi, W.R. Lu, Meromorphic functions that share two sets II, Acta Math. Sci. Ser. B, Eng Ed., 24 (2004), №1, 83–90.

	Some results on uniqueness of meromorphic functions sharing sets
Author	A. Sarkar arindamsarkarmath@gmail.com Department of Mathematics, Kandi Raj College, West Bengal, India
Abstract	Let f and g be two non constant meromorphic functions defined in the open complex plane C. In 2011 A. Banerjee ([4]), in an attempt to answer on the question of F. Gross ([9]), by W. C. Lin and H. X. Yi ([18]), improved a result of I. Lahiri ([14]) by reducing the cardinality of the set shared by f and g from 7 to 6 under weaker condition on ramification index. In this paper we show that the cardinality of the shared set can further be reduced to 4 as well as the condition on ramification index can be replaced by weaker one to obtain the same conclusion as A. Banerjee ([4]).
Keywords	meromorphic functions; weighted sharing; uniqueness
Reference	1. A. Banerjee, Uniqueness of meromorphic functions that share two sets, Southeast Asian Bull. Math., 31 (2007), 7–17. 2. A. Banerjee, On uniqueness of meromorphic functions that share two sets, Georgian Math. J., 15 (2008), №1, 21–28. 3. A. Banerjee, S. Mukherjee, Uniqueness of meromorphic functions sharing two or three sets, Hokkaido Math. J., 37 (2008), №3, 507–530. 4. A. Banerjee, A uniqueness result on meromorphic functions sharing two sets, J. Adv. Math. Stud., 4 (2011), №1, 1–10. 5. M. Fang, H. Guo, On meromorphic functions sharing two values, Analysis, 17 (1997), №4, 355–366. 6. M. Fang, I. Lahiri, Uniquene range set for certain meromorphic functions, Indian J. Math., 45 (2003), №2, 141–150. 7. M. Fang, W. Xu, A note on a problem of Gross, Chinese Ann. Math. Ser. A, 18 (1997), №4, 355–366. 8. G. Frank, M.R. Reinders, A unique range set for meromorphic functions with 11 elements, Complex Var. Theory Appl., 37 (1998), 185–193. 9. F. Gross, Factorization of meromorphic functions and some open problems, Proc. Conf. Univ. Kentucky, Lexington, Lecture notes in Math., Springer, Berlin, 599 (1977), 51–69. 10. Q. Han, H.X. Yi, Some further results on meromorphic functions that share two sets, Ann. Polon. Math. 91 (2008), №1, 17–31. 11. W.K. Hayman, Meromorphic functions. – Clarendon Press, Oxford, 1964. 12. I. Lahiri, Weighted sharing and uniqueness of meromorphic functions, Nagoya Math. J., 161 (2001), 193–206. 13. I. Lahiri, Weighted value sharing and uniqueness of meromorphic functions, Complex Var. Theory Appl., 46 (2001), 241–253. 14. I. Lahiri, On a question of Hong Xun Yi, Arch Math. (Brno), 38 (2002), 119–128. 15. I. Lahiri, S. Dewan, Value distribution of the product of a meromorphic function and its derivative, Kodai Math. J., 26 (2003), 95–100. 16. P. Li, C.C. Yang, On the unique range sets for meromorphic functions, Proc. Amer. Math. Soc., 124 (1996), 177–185. 17. P. Li, C.C. Yang, Some further results on the unique range sets for meromorphic functions, Kodai Math. J., 18 (1995), 437–450. 18. W.C. Lin, H.X. Yi, Some further results on meromorphic functions that share two sets, Kyungpook Math. J., 43 (2003), 73–85. 19. A.Z. Mohon’ko, On the Nevanlinna characteristics of some meromorphic functions, Theory of Funct. Funct. Anal. Appl., 14 (1971), 83–87. 20. H.X. Yi, Uniqueness of meromorphic functions and a question of Gross, Science in China(A), 37 (1994), №7, 802–813. 21. H.X. Yi, Unicity theorem for meromorphic functions or entire functions, Bull. Austral. Math. Soc. II, 52 (1995), 215–224. 22. H.X. Yi, On a question of Gross concerning uniqueness of entire functions, Bull. Austral. Math. Soc., 57 (1998), 343–349. 23. H.X. Yi, Meromorphic functions that share two sets (Chinese), Acta Math. Sinica (Chinese Ser.), 45 (2002), №1, 75–82. 24. H.X. Yi, Uniqueness of meromorphic functions II, Indian J. Pure Appl. Math., 28 (1997), 509–519. 25. H.X. Yi, W.C. Lin, Uniqueness of meromorphic and a question of Gross, Kyungpook Math. J., 46 (2006), 437–444. 26. H.X. Yi, W.R. Lu, Meromorphic functions that share two sets II, Acta Math. Sci. Ser. B, Eng Ed., 24 (2004), №1, 83–90.
Pages	53-61
Volume	41
Issue	1
Year	2014
Journal	Matematychni Studii
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Table of content of issue	html