A survey on different variants of shift operators of mermorphic function normally or partially sharing values or small functions
Main Article Content
Abstract
Qi [23] first demonstrated that if an entire function and its shift operator share two distinct periodic functions, then they must coincide. This foundational result has stimulated further research into analogous problems within broader and more generalized framework. In this survey, we systematically develop the theory of periodicity and related topics for various shift operators associated with meromorphic functions, under both normal and partial sharing conditions. This work offers a fresh perspective on the theory of shift operators, streamlining existing results to present them in a compact form. It also includes our own contributions to this specific area of literature with a number illustrative examples.
Keywords
Meromorphic functions; CM sharing; linear shift operator; linear differential polynomial; wider sense
Article Details
References
1. A. Banerjee and J. Banerjee, Normal and differentiable periodicity of linear shift operators under partial sharing, Siberian Elec. Math. Reports, (Accepted for publication).
2. A. Banerjee and S. Bhattacharyya, Uniqueness of meromorphic functions with their reduced linear c-shift operators sharing two or more values or sets, Adv. Di er. Equ., 509(2019),1-23.
3. A. Banerjee and S. Maity, Meromorphic function partially shares small functions or values with its linear c-shift operator, Bull. Korean Math. Soc., 58(5)(2021), 1175–1192.
4. A. Banerjee and A. Roy, Uniqueness of meromorphic functions ordinarily and partially sharing values with reduced linear c-shift operators, J. Indian Math. Soc., 88(3-4)(2021),201-216.
5. K. S. Charak, R. J. Korhonen, and G. Kumar, A note on partial sharing of values of meromorphic functions with their shifts, J. Math. Anal. Appl. 435(2)(2016), 1241-1248.
6. S. J. Chen, On uniqueness of meromorphic functions and their difference operators with partially shared values, Comput. Methods Funct. Theo., 18(2018), 529-536.
7. W. J. Chen and Z. G. Huang, Uniqueness of meromorphic functions concerning their derivatives and shifts with partially shared values, J. Contemp. Math. Anal., 57(4)(2022), 232–241.
8. Z. X. Chen and H. X. Yi, On sharing values of meromorphic functions and their differences, Results. Math., 63(1)(2013), 557–565.
9. N. Cui and Z. Chen, The conjecture on unity of meromorphic functions concerning their differences, J. Diff. Equ. Appl., 22(10)(2016), 1452-1471.
10. B. Deng, M. L. Fang and D. Liu, Unicity of mermorphic functions concerning shared functions with their difference, Bull. Korean Math. Soc., 56(6)(2019), 1511-1524.
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13. Y. Jiang and Z. Chen, Meromorphic functions share two values with its difference operator, An. Stiint. Univ.‘Al. I. Cuza’Iasi, 63(1)(2017), 169-175.
14. I. Kaish and R. Mondal, Meromorphic functions sharing values partially with their derivatives and shifts, J. Anal., 31(1)(2023), 329–342.
15. I. Kaish and R. Mondal, On a uniqueness question of meromorphic functions and partial shared value,, Com. Korean Math. Soc., 39(1)(2024), 105–116.
16. S. Li and Z. Gao, Entire functions sharing one or two finite values CM with their shifts or difference operators, Archiv der Math., 97(5)(2011), 475-483.
17. X. M. Li, H. X. Yi and C. Y. Kang, Results on meromorphic functions sharing three values with their difference operators, Bull. Korean Math. Soc., 52(5)(2015), 1401-1422.
18. S. Li, D. Mei and B. Chen, Uniqueness of meromorphic functions sharing two values with their difference operator, Adv. Differ. Equ., 390(2017), 1-9.
19. D. Liu, D. Yang and M. L. Fang, Unicity of Entire Functions concerning Shifts and Difference Operators, Abstra. Appl. Anal., 2014(2014), 1-5.
20. F. L¨u and W. L¨u, Meromorphic functions sharing three values with their difference operators, Comput. Methods Funct. Theo., 17(3)(2017), 395-403.
21. V. Noulorvang and D. T. Pham, On partial value sharing results of meromorphic functions with their shifts and its applications, Bull. Korean Math. Soc., 57(5)(2020), 1083–1094.
22. D.C. Pramanik and A. Sarkar, Uniqueness of shift and derivatives of meromorphic functions, Matematychni Studii., 61(2)(2024), 160–167.
23. D.C. Pramanik and N. Dey, Uniqueness of shift and derivative of a meromorphic function, J. Indian Math. Soc., 92(4)(2025), 617-626.
24. X. G. Qi, Value distribution and uniqueness of difference polynomials and entire solutions of difference equations, Ann Polon Math., 102(2)(2011), 129–142.
25. X. G. Qi and L. Z. Yang, Meromorphic functions that share values with their shifts or their n-th order differences, Anal. Math., 46(4)(2020), 843-865.
26. X. G. Qi and L. Z. Yang, Uniqueness of meromorphic functions concerning their shifts and derivatives, Comput. Methods Funct. Theo., 20(1)(2020), 159–178.
27. A. Roy and A. Banerjee, Linear delay-differential operator of a meromorphic function sharing two sets or small function together with values with its c-shift or q-shift., Stud. Univ. Babes-Bolyai Math., 68(2023)(3), 593–612.
28. J. Zhang and L. W. Liao, Entire functions sharing some values with their difference operators, Sci. China Math., 57(2014), 2143-2152.
2. A. Banerjee and S. Bhattacharyya, Uniqueness of meromorphic functions with their reduced linear c-shift operators sharing two or more values or sets, Adv. Di er. Equ., 509(2019),1-23.
3. A. Banerjee and S. Maity, Meromorphic function partially shares small functions or values with its linear c-shift operator, Bull. Korean Math. Soc., 58(5)(2021), 1175–1192.
4. A. Banerjee and A. Roy, Uniqueness of meromorphic functions ordinarily and partially sharing values with reduced linear c-shift operators, J. Indian Math. Soc., 88(3-4)(2021),201-216.
5. K. S. Charak, R. J. Korhonen, and G. Kumar, A note on partial sharing of values of meromorphic functions with their shifts, J. Math. Anal. Appl. 435(2)(2016), 1241-1248.
6. S. J. Chen, On uniqueness of meromorphic functions and their difference operators with partially shared values, Comput. Methods Funct. Theo., 18(2018), 529-536.
7. W. J. Chen and Z. G. Huang, Uniqueness of meromorphic functions concerning their derivatives and shifts with partially shared values, J. Contemp. Math. Anal., 57(4)(2022), 232–241.
8. Z. X. Chen and H. X. Yi, On sharing values of meromorphic functions and their differences, Results. Math., 63(1)(2013), 557–565.
9. N. Cui and Z. Chen, The conjecture on unity of meromorphic functions concerning their differences, J. Diff. Equ. Appl., 22(10)(2016), 1452-1471.
10. B. Deng, M. L. Fang and D. Liu, Unicity of mermorphic functions concerning shared functions with their difference, Bull. Korean Math. Soc., 56(6)(2019), 1511-1524.
11. W. K. Hayman, Meromorphic functions, The Clarendon Press, Oxford (1964).
12. J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo and J. Zhang, Value sharing results for shifts of meromorphic function and sufficient conditions for periodicity, J. Math. Anal. Appl., 355(2009), 352-363.
13. Y. Jiang and Z. Chen, Meromorphic functions share two values with its difference operator, An. Stiint. Univ.‘Al. I. Cuza’Iasi, 63(1)(2017), 169-175.
14. I. Kaish and R. Mondal, Meromorphic functions sharing values partially with their derivatives and shifts, J. Anal., 31(1)(2023), 329–342.
15. I. Kaish and R. Mondal, On a uniqueness question of meromorphic functions and partial shared value,, Com. Korean Math. Soc., 39(1)(2024), 105–116.
16. S. Li and Z. Gao, Entire functions sharing one or two finite values CM with their shifts or difference operators, Archiv der Math., 97(5)(2011), 475-483.
17. X. M. Li, H. X. Yi and C. Y. Kang, Results on meromorphic functions sharing three values with their difference operators, Bull. Korean Math. Soc., 52(5)(2015), 1401-1422.
18. S. Li, D. Mei and B. Chen, Uniqueness of meromorphic functions sharing two values with their difference operator, Adv. Differ. Equ., 390(2017), 1-9.
19. D. Liu, D. Yang and M. L. Fang, Unicity of Entire Functions concerning Shifts and Difference Operators, Abstra. Appl. Anal., 2014(2014), 1-5.
20. F. L¨u and W. L¨u, Meromorphic functions sharing three values with their difference operators, Comput. Methods Funct. Theo., 17(3)(2017), 395-403.
21. V. Noulorvang and D. T. Pham, On partial value sharing results of meromorphic functions with their shifts and its applications, Bull. Korean Math. Soc., 57(5)(2020), 1083–1094.
22. D.C. Pramanik and A. Sarkar, Uniqueness of shift and derivatives of meromorphic functions, Matematychni Studii., 61(2)(2024), 160–167.
23. D.C. Pramanik and N. Dey, Uniqueness of shift and derivative of a meromorphic function, J. Indian Math. Soc., 92(4)(2025), 617-626.
24. X. G. Qi, Value distribution and uniqueness of difference polynomials and entire solutions of difference equations, Ann Polon Math., 102(2)(2011), 129–142.
25. X. G. Qi and L. Z. Yang, Meromorphic functions that share values with their shifts or their n-th order differences, Anal. Math., 46(4)(2020), 843-865.
26. X. G. Qi and L. Z. Yang, Uniqueness of meromorphic functions concerning their shifts and derivatives, Comput. Methods Funct. Theo., 20(1)(2020), 159–178.
27. A. Roy and A. Banerjee, Linear delay-differential operator of a meromorphic function sharing two sets or small function together with values with its c-shift or q-shift., Stud. Univ. Babes-Bolyai Math., 68(2023)(3), 593–612.
28. J. Zhang and L. W. Liao, Entire functions sharing some values with their difference operators, Sci. China Math., 57(2014), 2143-2152.