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Meromorphic function partially shares small functions or values with its linear $c$-shift operator

Abhijit Banerjee|Sayantan Maity
대한수학회 (The Korean Mathematical Society)|2021.09|대한수학회보 (Bulletin of the KMS)|제58권 제5호|pp. 1175-1192|DOI 10.4134/BKMS.b200840

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키워드

uniquenessMeromorphic functionsmall functionlinear $c$-shift operatorpartial sharing
주제분류수학

초록

In this paper, we have studied on the uniqueness problems of meromorphic functions with its linear $c$-shift operator in the light of partial sharing. Our two results improve and generalize two very recent results of Noulorvang-Pham [Bull. Korean Math. Soc. 57 (2020), no. 5, 1083--1094] in some sense. In addition, our other results have improved and generalized a series of results due to L\"u-L\"u [Comput. Methods Funct. Theo. 17 (2017), no. 3, 395--403], Zhen [J. Contemp. Math. Anal. 54 (2019), no. 5, 296--301] and Banerjee-Bhattacharyya [Adv. Differ. Equ. 509 (2019), 1--23]. We have exhibited a number of examples to show that some conditions used in our results are essential.

Abstract

In this paper, we have studied on the uniqueness problems of meromorphic functions with its linear $c$-shift operator in the light of partial sharing. Our two results improve and generalize two very recent results of Noulorvang-Pham [Bull. Korean Math. Soc. 57 (2020), no. 5, 1083--1094] in some sense. In addition, our other results have improved and generalized a series of results due to L\"u-L\"u [Comput. Methods Funct. Theo. 17 (2017), no. 3, 395--403], Zhen [J. Contemp. Math. Anal. 54 (2019), no. 5, 296--301] and Banerjee-Bhattacharyya [Adv. Differ. Equ. 509 (2019), 1--23]. We have exhibited a number of examples to show that some conditions used in our results are essential.

참고문헌 (23)

  1. Abhijit Banerjee, "Uniqueness of meromorphic functions with their reduced linear c-shift operators sharing two or more values or sets", Advances in Difference Equations, 2019(1), pp.23~ (2019), Springer Science and Business Media LLC[DOI]
  2. K.S. Charak, "A note on partial sharing of values of meromorphic functions with their shifts", Journal of Mathematical Analysis and Applications, 435(2), pp.1241~1248 (2016), Elsevier BV[DOI]
  3. Chuang-Xin Chen, "A note on entire functions and their differences", Journal of Inequalities and Applications, 2013(1), pp.587~ (2013), Springer Science and Business Media LLC[DOI]
  4. Zong-Xuan Chen, "On Sharing Values of Meromorphic Functions and Their Differences", Results in Mathematics, 63(1-2), pp.557~565 (2013), Springer Science and Business Media LLC[DOI]
  5. Yik-Man Chiang, "On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane", The Ramanujan Journal, 16(1), pp.105~129 (2008), Springer Science and Business Media LLC[DOI]
  6. R.G. Halburd, "Difference analogue of the Lemma on the Logarithmic Derivative with applications to difference equations", Journal of Mathematical Analysis and Applications, 314(2), pp.477~487 (2006), Elsevier BV[DOI]
  7. R. G. Halburd, "Nevanlinna theory for the difference operator", Ann. Acad. Sci. Fenn. Math, 31(2), pp.463~478 (2006)
  8. R. G. Halburd, "Value distribution and linear operators", Proc. Edinb. Math. Soc, 57, pp.493~504 (2014)
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  10. W. K. Hayman, "Meromorphic Functions, Oxford Mathematical Monographs" (1964), Clarendon Press
  11. J. Heittokangas, "Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity", Journal of Mathematical Analysis and Applications, 355(1), pp.352~363 (2009), Elsevier BV[DOI]
  12. J. Heittokangas, "Uniqueness of meromorphic functions sharing values with their shifts", Complex Variables and Elliptic Equations, 56(1-4), pp.81~92 (2011), Informa UK Limited[DOI]
  13. Zhibo Huang, "Value distribution and uniqueness on q-differences of meromorphic functions", 대한수학회보, 50(4), pp.1157~1171 (2013), 대한수학회
  14. Z. -B. Huang, "Uniqueness of the Differences of Meromorphic Functions", Analysis Mathematica, 44(4), pp.461~473 (2018), Springer Science and Business Media LLC[DOI]
  15. Y. Jiang, "Meromorphic functions share two values with its difference operator", An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si. Mat. (N. S. ), 63(1), pp.169~175 (2017)
  16. Xiao-Min Li, "Notes on entire functions sharing an entire function of a smaller order with their difference operators", Archiv der Mathematik, 99(3), pp.261~270 (2012), Springer Science and Business Media LLC[DOI]
  17. Feng Lü, "Meromorphic Functions Sharing Three Values with their Difference Operators", Computational Methods and Function Theory, 17(3), pp.395~403 (2017), Springer Science and Business Media LLC[DOI]
  18. Xudan Luo, "Value sharing results for shifts of meromorphic functions", Journal of Mathematical Analysis and Applications, 377(2), pp.441~449 (2011), Elsevier BV[DOI]
  19. Vangty Noulorvang, "On partial value sharing results of meromorphic functions with their shifts and its applications", 대한수학회보, 57(5), pp.1083~1094 (2020), 대한수학회[DOI]
  20. Katsutoshi Yamanoi, "The second main theorem for small functions and related problems", Acta Mathematica, 192(2), pp.225~294 (2004), International Press of Boston[DOI]
  21. C. -C. Yang, "Uniqueness Theory of Meromorphic Functions" (2003), Kluwer Academic Publishers Group
  22. Jie Zhang, "Entire functions sharing some values with their difference operators", Science China Mathematics, 57(10), pp.2143~2152 (2014), Springer Science and Business Media LLC[DOI]
  23. L. Zhen, "Meromorphic functions sharing three polynomials with their difference operators", J. Contemp. Math. Anal, 54(5), pp.296~301 (2019)

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