Meromorphic function sharing three sets with its shift, q-shift,
q-difference and analogous operators

A. Banerjee; S. Bhattacharyya

doi:10.30970/ms.52.1.38-47

Using the notion of weighted sharing of three sets we deal with the uniqueness problem of meromorphic function with its shift, q-shift, q-difference and analogous operator. A handful number of examples have been provided to substantiate our certain claims. In other words, we deduce conditions providing a) $f(z) \equiv t f(z+c)$, where $t^n=1$, $n\ge 2$; b) $f(z)\equiv t f(qz+c)$, where $t^n=1$, $n\ge 2$ and $q , c \in \mathbb{C} \setminus \{0\}$; c) $f(z)\equiv tf(qz)$, where $t^n=1$ and $\mid q \mid =1$; d) $f(z)\equiv \chi_{_n}\Delta_qf$ for $n\ge 2$ with $\Delta_qf=f(qz)-f(z)$ and $\chi_{_n}= \begin{cases} 1, & \mbox{if } n=2, \\ t, & \mbox{if } n \geq 3,\; t^n=1. \end{cases}$

1. A. Banerjee, S. Mukherjee, Weighted sharing of three sets, Southeast Asian Bull. Math., 34 (2010), 11–23.

2. D.C. Barnett, R.G. Halburd, R.J. Korhonen, W. Morgan, Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations, Proc. Roy. Soc. Edinburgh Sect., A, 137 (2007), no.3, 457–474.

3. Y.M. Chian, S.J. Feng, On the Nevanlinna Characteristic f(z+) and difference equations in complex plane, Ramanujan J., 16 (2008), 105–129.

4. F. Gross, C.F. Osgood, Entire functions with common preimages, in: Factorization theory of meromorphic functions and related topics, Lecture Notes in Pure and Appl. Math., Marcel Dekker, New York, 1982, 19–24.

5. R.G. Halburd, R.J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl., 314 (2006), 477–487.

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

7. I. Lahiri, Value distribution of certain differential polynomials, Int. J. Math. Math. Sci., 28 (2001), no.2, 83–91.

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

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

10. X. Qi, L. Yang, Sets and value sharing of q-differences of meromorphic functions, Bull. Korean Math. Soc., 50 (2013), no.3, 731–745.

11. X. Qi, L. Yang, Sharing sets of q-difference of meromorphic functions, Math. Slovaca, 64 (2014), no.1, 51–60.

12. J.F. Xu, X.B. Zhang, The zeros of q-difference polynomials of meromorphic functions, Adv. Diff. Equ., 2012, 2012:200, 1–10.

13. C.C. Yang, On deficiencies of differential polynomials, II, Math. Z., 125 (1972), 107–112.

14. H.X. Yi, Meromorphic functions that share three sets, Kodai Math. J., 20 (1997), 22–32.

15. H.X. Yi, C.C. Yang, Theory of the Uniqueness of Meromorphic Functions, Science Press, Beijing, 1995.

16. J.L. Zhang, Value distribution and shared sets of differences of meromorphic functions, J. Math. Anal. Appl., 367 (2010), no.2, 401–408.

17. J.L. Zhang, R. Korhonen, On the Nevanlinna characteristic of f(qz) and its applications, J. Math. Anal. Appl., 369 (2010), 537–544.

	Meromorphic function sharing three sets with its shift, q-shift, q-difference and analogous operators
Author	A. Banerjee¹, S. Bhattacharyya² abanerjee kal@yahoo.co.in, abanerjeekal@gmail.com¹, saikat352@gmail.com, saikatbh89@yahoo.com² 1) Department of Mathematics, University of Kalyani West Bengal, India; 2) Department of Mathematics, University of Kalyani West Bengal, India
Abstract	Using the notion of weighted sharing of three sets we deal with the uniqueness problem of meromorphic function with its shift, q-shift, q-difference and analogous operator. A handful number of examples have been provided to substantiate our certain claims. In other words, we deduce conditions providing a) $f(z) \equiv t f(z+c)$, where $t^n=1$, $n\ge 2$; b) $f(z)\equiv t f(qz+c)$, where $t^n=1$, $n\ge 2$ and $q , c \in \mathbb{C} \setminus \{0\}$; c) $f(z)\equiv tf(qz)$, where $t^n=1$ and $\mid q \mid =1$; d) $f(z)\equiv \chi_{_n}\Delta_qf$ for $n\ge 2$ with $\Delta_qf=f(qz)-f(z)$ and $\chi_{_n}= \begin{cases} 1, & \mbox{if } n=2, \\ t, & \mbox{if } n \geq 3,\; t^n=1. \end{cases}$
Keywords	meromorphic functions; uniqueness; shift; difference operator; shared set
DOI	doi:10.30970/ms.52.1.38-47
Reference	1. A. Banerjee, S. Mukherjee, Weighted sharing of three sets, Southeast Asian Bull. Math., 34 (2010), 11–23. 2. D.C. Barnett, R.G. Halburd, R.J. Korhonen, W. Morgan, Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations, Proc. Roy. Soc. Edinburgh Sect., A, 137 (2007), no.3, 457–474. 3. Y.M. Chian, S.J. Feng, On the Nevanlinna Characteristic f(z+) and difference equations in complex plane, Ramanujan J., 16 (2008), 105–129. 4. F. Gross, C.F. Osgood, Entire functions with common preimages, in: Factorization theory of meromorphic functions and related topics, Lecture Notes in Pure and Appl. Math., Marcel Dekker, New York, 1982, 19–24. 5. R.G. Halburd, R.J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl., 314 (2006), 477–487. 6. W.K. Hayman, Meromorphic Functions, The Clarendon Press, Oxford, 1964. 7. I. Lahiri, Value distribution of certain differential polynomials, Int. J. Math. Math. Sci., 28 (2001), no.2, 83–91. 8. I. Lahiri, Weighted sharing and uniqueness of meromorphic functions, Nagoya Math. J., 161 (2001), 193–206. 9. I. Lahiri, Weighted value sharing and uniqueness of meromorphic functions, Complex Var. Theo. Appl., 46 (2001), 241–253. 10. X. Qi, L. Yang, Sets and value sharing of q-differences of meromorphic functions, Bull. Korean Math. Soc., 50 (2013), no.3, 731–745. 11. X. Qi, L. Yang, Sharing sets of q-difference of meromorphic functions, Math. Slovaca, 64 (2014), no.1, 51–60. 12. J.F. Xu, X.B. Zhang, The zeros of q-difference polynomials of meromorphic functions, Adv. Diff. Equ., 2012, 2012:200, 1–10. 13. C.C. Yang, On deficiencies of differential polynomials, II, Math. Z., 125 (1972), 107–112. 14. H.X. Yi, Meromorphic functions that share three sets, Kodai Math. J., 20 (1997), 22–32. 15. H.X. Yi, C.C. Yang, Theory of the Uniqueness of Meromorphic Functions, Science Press, Beijing, 1995. 16. J.L. Zhang, Value distribution and shared sets of differences of meromorphic functions, J. Math. Anal. Appl., 367 (2010), no.2, 401–408. 17. J.L. Zhang, R. Korhonen, On the Nevanlinna characteristic of f(qz) and its applications, J. Math. Anal. Appl., 369 (2010), 537–544.
Pages	38-47
Volume	52
Issue	1
Year	2019
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Meromorphic function sharing three sets with its shift, q-shift, q-difference and analogous operators