International Journal of Nonlinear Analysis and Applications

Some results on q-shift difference-differential polynomials sharing finite value

Document Type : Research Paper

Authors

1 Department of Mathematics, School of Engineering, Presidency University, Bangalore-560064, INDIA

2 Department of Mathematics, Government Science College, Chitraduga-577 501, India

Abstract

In this paper, we study the uniqueness of meromorphic functions with q-shift difference-differential polynomials $F=P(f)\prod\limits_{j=1}^{d}f(q_{j}z+c_{j})^{v_{j}}]^{(k)}$ and $G=[P(g)\prod\limits_{j=1}^{d}g(q_{j}z+c_{j})^{v_{j}}]^{(k)}$, where $P(z)$ is a non-constant polynomial with degree $n$ sharing a finite value. The results of this paper are an extension of the previous theorems given by Harina P. Waghamore and Rajeshwari S [19].

Keywords

[1] A. Banerjee, Meromorphic functions sharing one value, Int. J. Math. Math. Sci. 22 (2005), 3587–3598.
[2] A. Banerjee and M. Basir Ahamed, Results on meromorphic function sharing two sets with its linear-difference operator, J. Contemp. Math. Anal. 55 (2020), 143–155.
[3] C.N. Chaithra, S.H. Naveenkumar, and H.R. Jayarama, Further results about the transcendental meromorphic solution of a special Fermat-type equation, Int. J. Nonlinear Anal. Appl. (2023). 10.22075/ijnaa.2023.29809.4266
[4] G. Haldar, Some further q-shift difference results on Hayman conjecture, Rend. Circ. Mate. Palermo Ser. 2 71 (2022), 887–907.
[5] R.G. Halburd and 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] R.G. Halburd and R.J. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 2, 463–478.
[7] W.K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. Math. 70 (1959), 9–42.
[8] W.K. Hayman, Meromorphic Functions, Vol. 78, Clarendon Press, Oxford, 1964.
[9] H.R. Jayarama, S.H. Naveenkumar, and C.N. Chaithra, Uniqueness of certain differential polynomials with finite weight, J. Fractional Calculus Appl. 14 (2023), 1–13.
[10] I. Lahiri and S. Dewan, Value distribution of the product of a meromorphic function and its derivative, Kodai Math. J. 26 (2003), 95–100.
[11] P. Li and C.-C. Yang, Some further results on the unique range sets of meromorphic functions, Kodai Math. J. 18 (1995), 437–450.
[12] K. Liu, X.-L. Liu, and T.-B. Cao, Uniqueness and zeros of q-shift difference polynomials, Proc. Math. Sci. 121 (2011), 301–310.
[13] K. Liu, X.-L. Liu, and T.-B. Cao, Some results on zeros distributions and uniqueness of derivatives of difference polynomials, Ann. Polonici Math. 109 (2012), 137–150.
[14] K. Liu and X.-G. Qi, Meromorphic solutions of q-shift difference equations, Ann. Polonici Math. 101 (2011), 215–225.
[15] S. Mallick and M. Basir Ahamed, On uniqueness of a meromorphic function and its higher difference operators sharing two sets, Anal. Math. Phys. 12 (2022), 78.
[16] S.H. Naveenkumar, C.N. Chaithra, and H.R. Jayarama, On the transcendental solution of the Fermat type q-shift equation, Electronic J. Math. Anal. Appl. 11 (2023), 1-7.
[17] P. Sahoo and H. Karmakar, Uniqueness results related to certain q-shift difference polynomials, Ann. Alexandru Ioan Cuza Univ. Math. 2 (2018), 389–403.
[18] P. Sahoo and B. Saha, Some results on uniqueness of meromorphic functions sharing a polynomial, Tbilisi Math. J. 9 (2016), 59–70.
[19] H.P. Waghamore and S. Rajeshwari, Uniqueness of difference polynomials of meromorphic functions, Int. J. Pure Appl. Math. 107 (2016), 971–981.
[20] H.P. Waghamore and A. Tanuja, Uniqueness of difference polynomials of meromorphic functions, Bull. Calcutta Math. Soc. 105 (2013), no. 3, 227–236.
[21] J. Xu and H. Yi, Uniqueness of entire functions and differential polynomials, Bull. Korean Math. Soc. 18 (2007), 623–629.
[22] J. Xu and X. Zhang, The zeros of q-shift difference polynomials of meromorphic functions, Adv. Differ. Equ. 200 (2012), 1–10.
[23] L. Yang, Value Distribution Theory, Springer, 1993.
[24] C.-C. Yang and X. Hua, Uniqueness and value-sharing of meromorphic functions, Ann. Acad. Sci. Fen. Ser. A1 Math. 22 (1997), 395–406.
[25] C.C. Yang and H.X. Yi, Uniqueness Theory of Meromorphic Functions, Mathematics and its Applications, Dordrecht, Kluwer Academic Publishers Group, 2003.
[26] H.-X. Yi, Uniqueness Theory of Meromorphic Functions, Pure Appl. Math. Monographs, Science Press, 1995.
[27] J. Zhang and R. Korhonen, On the Nevanlinna characteristic of f(qz) and its applications, J. Math. Anal. Appl. 139 (2010), 5376–544.
[28] J. Zhang and L. Yang, Entire solutions of q-difference equations and value distribution of q-difference polynomials, Ann. Polonici Math. 109 (2013), 39–46.
International Journal of Nonlinear Analysis and Applications
Volume 15, Issue 10
October 2024
Pages 391-399
  • Receive Date: 26 January 2023
  • Accept Date: 30 September 2023