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Date Published:
23/06/2024
Online Published:
23/06/2024
Online Published:
21/05/2024
Abstract Views: 27
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Views PDF: 52
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Articles
How to Cite
Banerjee, A., & Banerjee, J. (2024). Weighted sharing of sets in wider sense under the ambit of higher indexed polynomials. Thang Long Journal of Science: Mathematics and Mathematical Sciences, 3(1). https://science.thanglong.edu.vn/index.php/volc/article/view/126
Weighted sharing of sets in wider sense under the ambit of higher indexed polynomials
Main Article Content
Abstract
In this article, with respect to the recently introduced definition of weighted sharing of sets in the wider sense [8], we have comprehensively
extended and improved some well known results in the literature. As the definition involves the manipulations of two polynomials, we perceive
that the characteristizations of the underlying polynomials become utmost important. This realization urges us to introduce the definition of the index
of polynomial. Noting that, in case of weighted sharing of sets the polynomials so far chosen are of lower index, after defining and streamlining
the index concept, we explore the influence of using polynomials with higher indices in the context of weighted sharing of sets in wider sense to
improve and extend a number of earlier results.
Article Details
Keywords
Meromorphic functions, polynomials, weighted sharing, shared sets, uniqueness
References
[1] A. Banerjee, Bi-unique range sets for meromorphic functions, Nihonkai Math. J., 24(2013), 121-134.
[2] A. Banerjee, On a question of Gross, J. Math. Anal. Appl., 327(2)(2007), 1273-1283.
[3] A. Banerjee, On the uniqueness of meromorphic functions that share two sets, Georgian Math. J., 15(1)(2008), 21-38.
[4] A. Banerjee, Uniqueness of meromorphic functions sharing two sets with finite weight, Port. Math. (N. S.), 65(1)(2008), 81-93.
[5] A. Banerjee, Uniqueness of meromorphic functions sharing two sets with finite weight II, Tamkang J. Math., 41(4)(2010), 379-392.
[6] A. Banerjee, Uniqueness of meromorphic function that share three sets II, Publ. Math. Debrecen., 76(1-2)(2010), 173-182.
[7] A. Banerjee, Some uniqueness results on meromorphic functions sharing three sets II, CUBO A Math. J., 13(3)(2011), 57-67.
[8] A. Banerjee and J. Banerjee, On strong uniqueness polynomials in the wider sense and associated unique range sets of the same category, Adv. Stud: Euro-Tbilisi Math. J., 16(4)(2023), 175-189.
[9] A. Banerjee and P. Bhattacharjee, Uniqueness and set sharing of derivatives of meromorphic functions, Math. Slovaca, 61(2)(2011), 197- 214.
[10] A. Banerjee and I. Lahiri, A uniqueness polynomial generating a uniquerange set and vise versa, Comput. Method Funct. Theo., 12(2)(2012), 527- 539.
[11] A. Banerjee and S. Mallick, Uniqueness properties of meromorphicfunctions in the light of three shared sets, Math. Slovaca, 68(3)(2018), 591-606.
[12] A. Banerjee and S. Mallick, On the characterisations of a new class of strong uniqueness polynomials generating unique range sets, Comput. Methods Funct. Theo., 17(2017), 19-45.
[13] A. Banerjee and S. Mukherjee, Uniqueness of meromorphic functions sharing two or three sets, Hokkaido Math. J., 37(3)(2008), 507-530.
[14] A. Banerjee and S. Mukherjee, Some uniqueness results on meromorphic functions sharing two or three sets, Sarajevo J. Math., 5(18)(2009), 169-189.
[15] G. Frank and M. Reinders, A unique range set for meromorphic functions with 11 elements, Com. Var. Theo. Appl., 37(1)(1998), 184-193.
[16] H. Fujimoto, On uniqueness of meromorphic functions sharing finite sets, Amer. J. Math., 122(6)(2000), 1175-1203.
[17] H. Fujimoto, On uniqueness polynomials for meromorphic functions, Nagoya Math. J., 170(2003), 33-46.
[18] F. Gross, Factorization of meromorphic functions and some open problems. Proc. Conf. Univ. Kentucky, Leixngton, Ky(1976); Lecture Notes in Math, 599(1977), 51-69, Springer(Berlin).
[19] W. K. Hayman, Meromorphic functions, The Clarendon Press, Oxford (1964).
[20] I. Lahiri, Weighted sharing and uniqueness of meromorphic functions, Nagoya Math. J., 161(2001), 193-206.
[21] I. Lahiri,Value distribution of certain differential polynomials, Int. J. Math. Math. Sci., 28(2)(2001), 83-91.
[22] I. Lahiri, Weighted value sharing and uniqueness of meromorphic functions, Complex Var. Theo. Appl., 46(3)(2001), 241-253.
[23] W. C. Lin and H. X. Yi, Some further results on meromorphic functions that share two sets, Kyungpook Math. J., 43(1)(2003), 73-85.
[24] . C. Lin and H. X. Yi, Uniqueness theorems for meromorphic functions that share three sets, Complex Var. Theory Appl., 48(2003), 315-327.
[25] W. C. Lin and H. X. Yi, Uniqueness of meromorphic functions and a question of Gross, Kyungpook Math. J., 46(3)(2006), 437-444.
[26] S. Mallick and A. Banerjee, Uniqueness and two shared set problems of meromorphic functions in a different angle, Math. Bohemica., 146(1)(2021), 19-45.
[27] H. X. Yi, Uniqueness of meromorphic functions and a question of gross, J. Sc. China(Ser A), 37(7)(1994).
[28] H. X. Yi, Unicity theorems for meromorphic or entire functions II, Bull. Austral. Math. Soc., 52(2)(1995), 215-224.
[29] H. X. Yi, Meromorphic functions that share one or two values II, Kodai Math. J., 22(2)(1999), 264-272.
[30] H. X. Yi and W. R. L¨u, Meromorphic functions that share two sets II, Acta Math. Sci. (Ser. B) Engl. Ed., 24(1)(2004), 83-90.
[31] A. D. Wu and W. C. Lin, Some properties of meromorphic functions concerning shared-sets, J. Classical Anal., 7(2015), 1-15.
[2] A. Banerjee, On a question of Gross, J. Math. Anal. Appl., 327(2)(2007), 1273-1283.
[3] A. Banerjee, On the uniqueness of meromorphic functions that share two sets, Georgian Math. J., 15(1)(2008), 21-38.
[4] A. Banerjee, Uniqueness of meromorphic functions sharing two sets with finite weight, Port. Math. (N. S.), 65(1)(2008), 81-93.
[5] A. Banerjee, Uniqueness of meromorphic functions sharing two sets with finite weight II, Tamkang J. Math., 41(4)(2010), 379-392.
[6] A. Banerjee, Uniqueness of meromorphic function that share three sets II, Publ. Math. Debrecen., 76(1-2)(2010), 173-182.
[7] A. Banerjee, Some uniqueness results on meromorphic functions sharing three sets II, CUBO A Math. J., 13(3)(2011), 57-67.
[8] A. Banerjee and J. Banerjee, On strong uniqueness polynomials in the wider sense and associated unique range sets of the same category, Adv. Stud: Euro-Tbilisi Math. J., 16(4)(2023), 175-189.
[9] A. Banerjee and P. Bhattacharjee, Uniqueness and set sharing of derivatives of meromorphic functions, Math. Slovaca, 61(2)(2011), 197- 214.
[10] A. Banerjee and I. Lahiri, A uniqueness polynomial generating a uniquerange set and vise versa, Comput. Method Funct. Theo., 12(2)(2012), 527- 539.
[11] A. Banerjee and S. Mallick, Uniqueness properties of meromorphicfunctions in the light of three shared sets, Math. Slovaca, 68(3)(2018), 591-606.
[12] A. Banerjee and S. Mallick, On the characterisations of a new class of strong uniqueness polynomials generating unique range sets, Comput. Methods Funct. Theo., 17(2017), 19-45.
[13] A. Banerjee and S. Mukherjee, Uniqueness of meromorphic functions sharing two or three sets, Hokkaido Math. J., 37(3)(2008), 507-530.
[14] A. Banerjee and S. Mukherjee, Some uniqueness results on meromorphic functions sharing two or three sets, Sarajevo J. Math., 5(18)(2009), 169-189.
[15] G. Frank and M. Reinders, A unique range set for meromorphic functions with 11 elements, Com. Var. Theo. Appl., 37(1)(1998), 184-193.
[16] H. Fujimoto, On uniqueness of meromorphic functions sharing finite sets, Amer. J. Math., 122(6)(2000), 1175-1203.
[17] H. Fujimoto, On uniqueness polynomials for meromorphic functions, Nagoya Math. J., 170(2003), 33-46.
[18] F. Gross, Factorization of meromorphic functions and some open problems. Proc. Conf. Univ. Kentucky, Leixngton, Ky(1976); Lecture Notes in Math, 599(1977), 51-69, Springer(Berlin).
[19] W. K. Hayman, Meromorphic functions, The Clarendon Press, Oxford (1964).
[20] I. Lahiri, Weighted sharing and uniqueness of meromorphic functions, Nagoya Math. J., 161(2001), 193-206.
[21] I. Lahiri,Value distribution of certain differential polynomials, Int. J. Math. Math. Sci., 28(2)(2001), 83-91.
[22] I. Lahiri, Weighted value sharing and uniqueness of meromorphic functions, Complex Var. Theo. Appl., 46(3)(2001), 241-253.
[23] W. C. Lin and H. X. Yi, Some further results on meromorphic functions that share two sets, Kyungpook Math. J., 43(1)(2003), 73-85.
[24] . C. Lin and H. X. Yi, Uniqueness theorems for meromorphic functions that share three sets, Complex Var. Theory Appl., 48(2003), 315-327.
[25] W. C. Lin and H. X. Yi, Uniqueness of meromorphic functions and a question of Gross, Kyungpook Math. J., 46(3)(2006), 437-444.
[26] S. Mallick and A. Banerjee, Uniqueness and two shared set problems of meromorphic functions in a different angle, Math. Bohemica., 146(1)(2021), 19-45.
[27] H. X. Yi, Uniqueness of meromorphic functions and a question of gross, J. Sc. China(Ser A), 37(7)(1994).
[28] H. X. Yi, Unicity theorems for meromorphic or entire functions II, Bull. Austral. Math. Soc., 52(2)(1995), 215-224.
[29] H. X. Yi, Meromorphic functions that share one or two values II, Kodai Math. J., 22(2)(1999), 264-272.
[30] H. X. Yi and W. R. L¨u, Meromorphic functions that share two sets II, Acta Math. Sci. (Ser. B) Engl. Ed., 24(1)(2004), 83-90.
[31] A. D. Wu and W. C. Lin, Some properties of meromorphic functions concerning shared-sets, J. Classical Anal., 7(2015), 1-15.