Non-linear difference polynomials sharing a polynomial with finite weight

Harina Pandit Waghamore, Preetham Nataraj Raj

Abstract


The uniqueness theory of meromorphic function mainly studies the conditions under which there exists only one function satisfying these conditions. The uniqueness theory of entire and meromorphic functions has grown up as an extensive sub-field of value distribution theory and the Nevanlinna's Five value and Four value theorems serves as the starting point of this uniqueness theory. In this paper, we consider a linear difference polynomial $\mathcal{L}_{\eta}(\mathfrak{f})=\mathfrak{f}(z+\eta)+\eta_0\mathfrak{f}(z)$, of the finite ordered non-constant meromorphic function $\mathfrak{f}$, with $\eta$ and $\eta_0$ being finite non-zero complex constants, and with the help of Nevanlinna theory, we analyse the uniqueness results between two finite ordered non-constant meromorphic functions $\mathfrak{f}$ and $\mathfrak{g}$, when their non-linear difference polynomials $\mathfrak{f}^n(z)\mathcal{L}_{\eta}(\mathfrak{f})$ and $\mathfrak{g}^n(z)\mathcal{L}_{\eta}(\mathfrak{g})$, with $n \ge 2$ being a positive integer shares a non-zero polynomial $p(z)$ with finite weights 0,1 and 2. Our results extend and improve some of the earlier results of Majumder (\textit{Applied Mathematics E-Notes, (17): 114-123, 2017)

Keywords


Meromorphic functions; difference polynomials; weighted sharing; uniqueness

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References


T. C. Alzahary and H. X. Yi. Weighted value sharing and a question of I. Lahiri. Complex Variables, Theory and Application: An International Journal, 49(15):1063–1078, 2004.

A. Banerjee. Meromorphic functions sharing one value. International Journal of Mathematics and Mathematical Sciences, 2005:3587–3598, 2005.

A. Banerjee and T. Biswas. On the value sharing of shift monomials of meromorphic functions. Surv. Math. Appl, 15:341–369, 2020.

Y.-M. Chiang and S.-J. Feng. On the Nevanlinna characteristic of f(z + ) and difference equations in the complex plane. The Ramanujan Journal, 16(1):105–129, 2008.

R. Halburd and R. Korhonen. Nevanlinna theory for the difference operator. arXiv preprint math/0506011, 2005.

R. Halburd and R. Korhonen. Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. Journal of Mathematical Analysis and Applications, 314(2):477–487, 2006.

G. Haldar. Uniqueness of entire functions whose difference polynomials share a polynomial with finite weight. Cubo (Temuco), 24(1):167–186, 2022.

W. Hayman. Meromorphic functions. Clarendon, 1964.

J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and J. Zhang. Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity. Journal of Mathematical Analysis and Applications, 355(1):352–363, 2009.

V. Husna, Veena, and S. Rajeshwari. Certain types of difference polynomials. Mathematics, 47:565–577, 2022.

I. Lahiri. Weighted sharing and uniqueness of meromorphic functions. Nagoya Mathematical Journal, 161:193–206, 2001.

I. Lahiri and S. Dewan. Value distribution of the product of a meromorphic function and its derivative. Kodai Mathematical Journal, 26(1):95–100, 2003.

I. Laine and C.-C. Yang. Value distribution of difference polynomials. Proc. Japan Acad. Ser. A Math. Sci., 83(8):148–151, 2007.

K. Liu and L.-Z. Yang. Value distribution of the difference operator. Archiv der Mathematik, 92(3), 2009.

K. Liu, X. Liu, and T. Cao. Value distributions and uniqueness of difference polynomials. Advances in Difference Equations, 2011:1–12, 2011.

Y. Liu, J. Wang, and F. Liu. Some results on value distribution of the difference operator. Bulletin of the Iranian Mathematical Society, 41(3):603–611, 2015.

S. Majumder. Uniqueness and value distribution of differences of meromorphic functions. Applied Mathematics E-Notes, 17:114–123, 2017.

P. N. Raj and H. P. Waghamore. Results on uniqueness of a polynomial and difference differential polynomial. Advanced Studies: Euro-Tbilisi Mathematical Journal, 16(2):79–96, 2023.

H. P. Waghamore and P. N. Raj. Uniqueness results on meromorphic functions concerning their shift and differential polynomial. Serdica Math. J, 47:191–212, 2021.

H. P. Waghamore and P. N. Raj. Uniqueness of q-difference of meromorphic functions sharing a small function with finite weight. Creative Mathematics & Informatics, 32(2), 2023.

C. Yang and H. Yi. Uniqueness theory of meromorphic functions, ser. Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers Group, 557, 2003.

C.-C. Yang. On deficiencies of differential polynomials, ii. Mathematische Zeitschrift, 125:107–112, 1972.

L. Yang. Value distribution theory. Springer-Verlag, Berlin; Science Press Beijing, Beijing, 1993. Translated and revised from the 1982 Chinese original.

C. Zongxuan, H. Zhibo, and Z. Xiumin. On properties of difference polynomials. Acta Mathematica Scientia, 31(2):627–633, 2011.




DOI: http://dx.doi.org/10.23755/rm.v51i0.1421

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