Uniqueness results on certain polynomials of meromorphic functions sharing a small function

Articles in Press

Document Type : Research Paper

Authors
Harina P Waghamore
Naveenkumar B. N.
Department of Mathematics, Bangalore University, Jnana Bharathi Campus, Bangalore -560 056, India
10.22075/ijnaa.2025.36923.5405
Abstract
This study explores specific polynomials that share a small function by applying the notions of weakly weighted and relaxed weighted sharing of meromorphic functions. In particular, the research delves into the uniqueness of two polynomial types related to the meromorphic function $f$: the homogeneous differential polynomial $\Phi[z]$ and the non-constant differential-difference polynomial $\Psi[z, f]$ and examines the value distribution of these polynomial functions within the context of weakly weighted and relaxed weighted sharing, resulting in the equation $\Phi[z] \equiv \Psi[z, f]$ (or) $\Phi[z]. \Psi[z, f] \equiv a^2$. The findings of this investigation enhance and extend the previous work of Harina P. Waghamore and Vijayalakshmi S. B. [27].
Keywords
Uniqueness
meromorphic function
homogeneous differential polynomial
Differential-difference polynomial
Weakly weighted sharing
Relaxed weighted sharing
[1] A.H. Al‑Khaladi, Notes on meromorphic functions sharing small function and its derivatives, Arab. J. Math. Sci. 21 (2015), no. 2, 194-208.
[2] A. Banerjee, Meromorphic functions sharing one value, Int. J. Math. Math. Sci. 22 (2005), 3587-3598.
[3] A. Banerjee and S. Mukherjee, Uniqueness of meromorphic functions concerning differential monomials sharing the same value, Bull. Math. Soc. Sci. 50 (2007), 191-206.
[4] A. Banerjee and S. Majumder, Some uniqueness results related to meromorphic function that share a small function with its derivative, Math. Rep. (Bucur.) 16 (2014), 95-111.
[5] A. Banerjee and S. Majumder, Uniqueness of certain type of differential‑difference and difference polynomials, Tamkang J. Math. 49 (2018), no. 1, 1-24.
[6] A. Banerjee and T. Biswas, On the value sharing of shift monomial of meromorphic functions, Surv. Math. Appl. 15 (2020), 341-369.
[7] K.S. Charak and B. Lal, Uniqueness of and , Turkish J. Math. 40 (2016), 569-581.
[8] R.G. Halburd and R.J. Korhonen, Finite‑order meromorphic solutions and the discrete Painleve equations, Proc. Lond. Math. Soc. 94 (2007), no. 2, 443-474.
[9] W.K. Hayman, Meromorphic Functions, Oxford Math. Monographs, Clarendon Press, Oxford, 1964.
[10] J. Heittokangas, R. Korhonen, I. Laine, and J. Rieppo, Uniqueness of meromorphic functions sharing values with their shifts, Complex Var. Elliptic Equ. 56 (2011), no. 1-4, 81-92.
[11] I. Lahiri, Weighted value sharing and uniqueness of meromorphic functions, Complex Var. Theory Appl. 46 (2001), 241-253.
[12] I. Lahiri and A. Sarkar, Uniqueness of a meromorphic function and its derivative, J. Inequal. Pure Appl. Math. 5 (2004), no. 1, 9.
[13] I. Lahiri and B. Pal, Uniqueness of meromorphic functions with their homogeneous and linear differential polynomials sharing a small function, Bull. Korean Math. Soc. 54 (2017), 825-838.
[14] I. Laine and C.C. Yang, Value distribution of difference polynomials, Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), no. 8, 148-151.
[15] S. Lin and W. Lin, Uniqueness of meromorphic functions concerning weakly weighted sharing, Kodai Math. J. 29 (2006), 269-280.
[16] X.Q. Lin and W.C. Lin, Uniqueness of entire functions sharing one value, Acta Math. Sci. Ser. B Engl. Ed. 31 (2011), 1062-1076.
[17] L. Liu and Y. Gu, Uniqueness of meromorphic functions that share one small function with their derivatives, Kodai Math. J. 27 (2004), no. 3, 272-279.
[18] K. Liu, X.L. Liu, and T.B. Cao, Some results on zeros and uniqueness of difference‑differential polynomials, Appl. Math. J. Chinese Univ. Ser. B 27 (2012), no. 1, 94-104.
[19] X. Luo and W.C. Lin, Value sharing results for shifts of meromorphic functions, J. Math. Anal. Appl. 377 (2011), no. 2, 441-449.
[20] B.N. NaveenKumar and H.P. Waghamore, Zeros and Weighted value sharing of q‑shift difference‑differential polynomials of entire and meromorphic functions, J. Fract. Calc. Appl. 15 (2024), no. 1, 1-20.
[21] B. Saha and T. Biswas, Uniqueness of entire functions concerning the shift differential polynomials, Electron. J. Math. Anal. Appl. 9 (2021), no. 2, 195-205.
[22] P. Sahoo and S. Seikh, Nonlinear differential polynomials sharing a small function, Mat. Vesnik 65 (2013), no. 252, 151-165.
[23] P. Sahoo and H. Karmakar, Results on uniqueness of entire functions whose difference polynomials share a polynomial, J. Contemp. Math. Anal. 52 (2017), no. 2, 102-110.
[24] H.P. Waghamore and V. Husna, Uniqueness of a meromorphic functions that share one small function and its derivative, Electron. J. Math. Anal. Appl. 4 (2016), no. 2, 25-37.
[25] H.P. Waghamore and S.H. Naveen Kumar, Results on uniqueness of meromorphic functions of differential polynomial, Malaya J. Mat. 6 (2018), 14-20.
[26] H.P. Waghamore and R. Maligi, Weighted sharing of Q‑Shift difference‑differential polynomials of meromorphic functions sharing a small function, Electron. J. Math. Anal. Appl. 7 (2019), no. 2, 366-378.
[27] H.P. Waghamore and S.B. Vijayalakmi, Results on uniqueness of differential monomial and differential polynomial, J. Appl. Math. Inform. 37 (2019), no. 3-4, 281-294.
[28] H.P. Waghamore and S.B. Vijayalakmi, Unicity result sharing one small function with the general differential‑difference polynomial, J. Anal. 29 (2021), no. 3, 781-801.
[29] H.P. Waghamore and B.N. Naveenkumar, Some Results on Uniqueness of Meromorphic Functions Regarding Shifts and Derivatives, Int. J. Differ. Equ. 18 (2023), no. 1, 251-265.
[30] H.P. Waghamore and B.N. Naveenkumar, Uniqueness results on difference product of entire functions, Electron. J. Math. Anal. Appl. 12 (2024), no. 1, 1-18.
[31] H.P. Waghamore and B.N. Naveenkumar, Uniqueness of Differential Polynomials sharing two values, Int. J. Open Probl. Comput. Math. 17 (2024), no. 3, 1-13.
[32] H.P. Waghamore and B.N. Naveenkumar, Uniqueness of certain polynomials of meromorphic functions sharing a set with finite weight, Serdica Math. J. 50 (2024), no. 3-4, 323-344.
[33] H.Y. Xu and Y. Hu, Uniqueness of meromorphic function and its differential polynomial concerning weakly weighted sharing, Gen. Math. 19 (2011), 101-111.
[34] C.C. Yang and H.X. Yi, Uniqueness theory of meromorphic functions, Math. Appl. 557, Kluwer Academic Publishers, Dordrecht, 2003.
[35] C.C. Yang and I. Laine, On analogies between nonlinear difference and differential equations, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), no. 1, 10-14.
[36] L. Yang, Value Distribution Theory, Springer‑Verlag, Berlin, 1993.
[37] L. Yang, Value distribution theory, translated and revised from the 1982 Chinese original, Springer‑Verlag, Berlin, 1993.
[38] H.X. Yi, Meromorphic functions that share one or two values II, Kodai Math. J. 22 (1999), no. 2, 264-272.
[39] K. Yu, On entire and meromorphic functions that share small functions with their derivatives, J. Inequal. Pure Appl. Math. 4 (2003), no. 1, 1-7.
[40] J. Zhang, Meromorphic functions sharing a small function with their derivatives, Kyungpook Math. J. 49 (2009), no. 1, 143-154.
[41] X.M. Zheng and Z.X. Chen, On deficiencies of some difference polynomials, Acta Math. Sin. (Chin. Ser.) 54 (2011), no. 6, 983-992.
[42] X.M. Zheng and H.Y. Xu, On the Deficiencies of Some Differential‑Difference Polynomials, Abstr. Appl. Anal. 2014 (2014), no. 1, 378151.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 29 June 2026

  • Receive Date 17 February 2025
  • Accept Date 03 April 2025