Unique range sets for powers of meromorphic functions

S. Mallick; D. Sarkar

doi:10.15330/ms.50.2.143-157

The prime concern of the paper is to deal with the notion of the unique range set for powers of meromorphic functions. As a consequence, we show that the lower bound of URSM (URSE) and URSM-IM (URSE-IM) can be significantly reduced up to cardinality $4$ $(4)$ from $11$ $(7)$ and $17$ $(10)$ respectively, for a class of power of meromorphic (entire) functions. Various applications of our main result also improve and generalize different results of Khoai-An-Lai (Internat. J. Math., 2018), Yi (Nagoya Math. J., 1995) and (J. Shandong Univ. Nat. Sci., 1998). Moreover, on the basis of some new notions introduced in the paper, our main result and its applications, we have partially reduced the known Gross Problem to a more narrow formulation and posed a number of open problems in the last section to unveil the least cardinality problem of unique range sets.

1. A. Banerjee, On the uniqueness of meromorphic functions that share two sets, Georgian Math. J., 15 (2008), №1, 1–18, DOI: 10.1515/GMJ.2008.21.

2. A. Banerjee, Uniqueness of meromorphic functions sharing two sets with finite weight II, Tamkang J. Math., 41 (2010), №4, 379–392. DOI: 10.5556/j.tkjm.41.2010.787.

3. A. Banerjee, Fujimoto’s theorem — a further study, J. Contemp. Math. Anal., 51 (2016), №4, 199–207. DOI: 10.3103/S1068362316040051.

4. A. Banerjee, I. Lahiri, A uniqueness polynomial generating a unique range set and vise versa, Comput. Methods Funct. Theory, 12 (2012), №2, 527–539. DOI: 10.1007/BF03321842.

5. A. Banerjee, S. Mallick, On the characterisations of a new class of strong uniqueness polynomials generating unique range sets, Comput. Methods Funct. Theory, 17 (2017), 19–45. DOI: 10.1007/s40315- 016-0174-y.

6. S. Bartels, Meromorphic functions sharing a set with 17 elements ignoring multiplicities, Complex Var. Elliptic Equ., 39 (1998), 85–92. DOI: 10.1080/17476939908815183.

7. M.L. Fang, H. Guo, On unique range sets for meromorphic or entire functions, Acta Math. Sinica (New Ser.), 14 (1998), №4, 569–576. DOI: 10.1007/BF02580416.

8. G. Frank, M. Reinders, A unique range set for meromorphic functions with 11 elements, Complex Var. Theory Appl., 37 (1998), №1, 185–193. DOI: 10.1080/17476939808815132.

9. H. Fujimoto, On uniqueness of meromorphic functions sharing finite sets, Amer. J. Math., 122 (2000), 1175.1203. DOI: 10.1353/ajm.2000.0045.

10. F. Gross, Factorization of meromorphic functions and some open problems, Proc. Conf. Univ. Kentucky, Leixngton, Kentucky (1976); Lecture Notes in Math., 599 (1977), 51.69, Springer (Berlin).

11. F. Gross, C.C. Yang, On preimage and range sets of meromorphic functions, Proc. Japan Acad. Ser. A Math. Sci., 58 (1982), №1, 17.20. DOI: 10.3792/pjaa.58-17.

12. W.K. Hayman, Meromorphic Functions, The Clarendon Press, Oxford, 1964.

13. H.H. Khoai, V.H. An, N.X. Lai, Strong uniqueness polynomials of degree 6 and unique range sets for powers of meromorphic functions, Internat. J. Math., 29 (2018), №5, 1850037-1 - 1850037-19. DOI: 10.1142/S0129167X18500374.

14. I. Lahiri, Value distribution of certain differential polynomials, Int. J. Math. Math. Sci., 28(2001), №2, 83-91. DOI: 10.1155/S0161171201011036.

15. I. Lahiri, Weighted sharing and uniqueness of meromorphic functions, Nagoya Math. J., 161 (2001), 193-206. DOI: 10.1017/S0027763000027215.

16. I. Lahiri, Weighted value sharing and uniqueness of meromorphic functions, Complex Var. Elliptic Equ., 46 (2001), 241-253. DOI: 10.1080/17476930108815411.

17. P. Li, C.C. Yang, Some further results on the unique range sets of meromorphic functions, Kodai Math. J., 18 (1995), 437-450. DOI: 10.2996/kmj/1138043482.

18. P. Li, C.C. Yang, On the unique range set of meromorphic functions, Proc. Amer. Math. Soc., 124(1996), №1, 177-185. DOI: 10.1090/S0002-9939-96-03045-6.

19. C.C. Yang, H.X. Yi, Uniqueness theory of meromorphic functions, Kluwer Academic Publishers, 2003.

20. H.X. Yi, A question of Gross and the uniqueness of entire functions, Nagoya Math. J., 138 (1995), 169-177. DOI: 10.1017/S0027763000005225.

21. H.X. Yi, Unicity theorems for meromorphic or entire functions III, Bull. Aust. Math. Soc., 53 (1996), 71-82. DOI: 10.1017/S0004972700016737.

22. H.X. Yi, The reduced unique range sets for entire or meromorphic functions, Complex Var. Elliptic Equ., 32 (1997), 191-198. DOI: 10.1080/17476939708814990.

23. H.X. Yi, The reduced unique range sets of meromorphic functions, J. Shandong Univ. Nat. Sci., 33 (1998), 361-368.

24. H.X. Yi, Meromorphic functions that share one or two values II, Kodai Math. J., 22 (1999), 264-272. DOI: 10.2996/kmj/1138044046.

25. H.X. Yi, W.R. Lu, Meromorphic functions that share two sets II, Acta Math. Sci. Ser. B Engl. Ed., 24 (2004), №1, 83-90. DOI: 10.1016/S0252-9602(17)30363-6.

	Unique range sets for powers of meromorphic functions
Author	S. Mallick¹, D. Sarkar² sanjay.mallick1986@gmail.com¹, smallick.ku@gmail.com² Department of Mathematics, Cooch Behar Panchanan Barma University, West Bengal, India
Abstract	The prime concern of the paper is to deal with the notion of the unique range set for powers of meromorphic functions. As a consequence, we show that the lower bound of URSM (URSE) and URSM-IM (URSE-IM) can be significantly reduced up to cardinality $4$ $(4)$ from $11$ $(7)$ and $17$ $(10)$ respectively, for a class of power of meromorphic (entire) functions. Various applications of our main result also improve and generalize different results of Khoai-An-Lai (Internat. J. Math., 2018), Yi (Nagoya Math. J., 1995) and (J. Shandong Univ. Nat. Sci., 1998). Moreover, on the basis of some new notions introduced in the paper, our main result and its applications, we have partially reduced the known Gross Problem to a more narrow formulation and posed a number of open problems in the last section to unveil the least cardinality problem of unique range sets.
Keywords	meromorphic function; entire function; uniqueness; unique range set; reduced unique range set
DOI	doi:10.15330/ms.50.2.143-157
Reference	1. A. Banerjee, On the uniqueness of meromorphic functions that share two sets, Georgian Math. J., 15 (2008), №1, 1–18, DOI: 10.1515/GMJ.2008.21. 2. A. Banerjee, Uniqueness of meromorphic functions sharing two sets with finite weight II, Tamkang J. Math., 41 (2010), №4, 379–392. DOI: 10.5556/j.tkjm.41.2010.787. 3. A. Banerjee, Fujimoto’s theorem — a further study, J. Contemp. Math. Anal., 51 (2016), №4, 199–207. DOI: 10.3103/S1068362316040051. 4. A. Banerjee, I. Lahiri, A uniqueness polynomial generating a unique range set and vise versa, Comput. Methods Funct. Theory, 12 (2012), №2, 527–539. DOI: 10.1007/BF03321842. 5. A. Banerjee, S. Mallick, On the characterisations of a new class of strong uniqueness polynomials generating unique range sets, Comput. Methods Funct. Theory, 17 (2017), 19–45. DOI: 10.1007/s40315- 016-0174-y. 6. S. Bartels, Meromorphic functions sharing a set with 17 elements ignoring multiplicities, Complex Var. Elliptic Equ., 39 (1998), 85–92. DOI: 10.1080/17476939908815183. 7. M.L. Fang, H. Guo, On unique range sets for meromorphic or entire functions, Acta Math. Sinica (New Ser.), 14 (1998), №4, 569–576. DOI: 10.1007/BF02580416. 8. G. Frank, M. Reinders, A unique range set for meromorphic functions with 11 elements, Complex Var. Theory Appl., 37 (1998), №1, 185–193. DOI: 10.1080/17476939808815132. 9. H. Fujimoto, On uniqueness of meromorphic functions sharing finite sets, Amer. J. Math., 122 (2000), 1175.1203. DOI: 10.1353/ajm.2000.0045. 10. F. Gross, Factorization of meromorphic functions and some open problems, Proc. Conf. Univ. Kentucky, Leixngton, Kentucky (1976); Lecture Notes in Math., 599 (1977), 51.69, Springer (Berlin). 11. F. Gross, C.C. Yang, On preimage and range sets of meromorphic functions, Proc. Japan Acad. Ser. A Math. Sci., 58 (1982), №1, 17.20. DOI: 10.3792/pjaa.58-17. 12. W.K. Hayman, Meromorphic Functions, The Clarendon Press, Oxford, 1964. 13. H.H. Khoai, V.H. An, N.X. Lai, Strong uniqueness polynomials of degree 6 and unique range sets for powers of meromorphic functions, Internat. J. Math., 29 (2018), №5, 1850037-1 - 1850037-19. DOI: 10.1142/S0129167X18500374. 14. I. Lahiri, Value distribution of certain differential polynomials, Int. J. Math. Math. Sci., 28(2001), №2, 83-91. DOI: 10.1155/S0161171201011036. 15. I. Lahiri, Weighted sharing and uniqueness of meromorphic functions, Nagoya Math. J., 161 (2001), 193-206. DOI: 10.1017/S0027763000027215. 16. I. Lahiri, Weighted value sharing and uniqueness of meromorphic functions, Complex Var. Elliptic Equ., 46 (2001), 241-253. DOI: 10.1080/17476930108815411. 17. P. Li, C.C. Yang, Some further results on the unique range sets of meromorphic functions, Kodai Math. J., 18 (1995), 437-450. DOI: 10.2996/kmj/1138043482. 18. P. Li, C.C. Yang, On the unique range set of meromorphic functions, Proc. Amer. Math. Soc., 124(1996), №1, 177-185. DOI: 10.1090/S0002-9939-96-03045-6. 19. C.C. Yang, H.X. Yi, Uniqueness theory of meromorphic functions, Kluwer Academic Publishers, 2003. 20. H.X. Yi, A question of Gross and the uniqueness of entire functions, Nagoya Math. J., 138 (1995), 169-177. DOI: 10.1017/S0027763000005225. 21. H.X. Yi, Unicity theorems for meromorphic or entire functions III, Bull. Aust. Math. Soc., 53 (1996), 71-82. DOI: 10.1017/S0004972700016737. 22. H.X. Yi, The reduced unique range sets for entire or meromorphic functions, Complex Var. Elliptic Equ., 32 (1997), 191-198. DOI: 10.1080/17476939708814990. 23. H.X. Yi, The reduced unique range sets of meromorphic functions, J. Shandong Univ. Nat. Sci., 33 (1998), 361-368. 24. H.X. Yi, Meromorphic functions that share one or two values II, Kodai Math. J., 22 (1999), 264-272. DOI: 10.2996/kmj/1138044046. 25. H.X. Yi, W.R. Lu, Meromorphic functions that share two sets II, Acta Math. Sci. Ser. B Engl. Ed., 24 (2004), №1, 83-90. DOI: 10.1016/S0252-9602(17)30363-6.
Pages	143-157
Volume	50
Issue	2
Year	2018
Journal	Matematychni Studii
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Table of content of issue	html