Uniqueness of entire functions whose difference polynomials share a polynomial with finite weight

Goutam Haldar

doi:10.4067/S0719-06462022000100167

DOI:

https://doi.org/10.4067/S0719-06462022000100167

Abstract

In this paper, we use the concept of weighted sharing of values to investigate the uniqueness results when two difference polynomials of entire functions share a nonzero polynomial with finite weight. Our result improves and extends some recent results due to Sahoo-Karmakar [J. Cont. Math. Anal. 52(2) (2017), 102–110] and that of Li et al. [Bull. Malays. Math. Sci. Soc., 39 (2016), 499–515]. Some examples have been exhibited which are relevant to the content of the paper.

Keywords

Entire function , difference polynomial , shift and difference operator , weighted sharing

Goutam Haldar Department of Mathematics, Malda College, Rabindra Avenue, Malda, West Bengal 732101, India.

Pages: 167–186
Date Published: 2022-04-12
Vol. 24 No. 1 (2022)

M. B. Ahamed, “An investigation on the conjecture of Chen and Yi”, Results Math., vol. 74, no. 3, paper No. 122, 28 pages, 2019.

T. C. Alzahary and H. X. Yi, “Weighted value sharing and a question of I. Lahiri”, Complex Var. Theory Appl., vol. 49, no. 15, pp. 1063–1078, 2004.

T. T. H. An and N. V. Phuong, “A lemma about meromorphic functions sharing a small function”, Comput. Methods Funct. Theory. doi: 10.1007/s40315-021-00388-3.

A. Banerjee, “Meromorphic functions sharing one value”, Int. J. Math. Math. Sci., no. 22, pp. 3587–3598, 2005.

A. Banerjee and S. Majumder, “Uniqueness of certain type of differential-difference and difference polynomials”, Tamkang J. Math., vol. 49, no. 1, pp. 1–24, 2018.

A. Banerjee and T. Biswas, “On the value sharing of shift monomial of meromorphic functions”, Surv. Math. Appl., vol. 15, pp. 341–369, 2020.

Z.-X. Chen, Z.-B. Huang and X.-M. Zheng, “On properties of difference polynomials”, Acta Math. Sci. Ser. B (Engl. Ed.), vol. 31, no. 2, pp. 627–633, 2011.

Y.-M. Chiang and S.-J. Feng, “On the Nevanlinna characteristic of f(z + Î·) and difference equations in the complex plane”, Ramanujan J., vol. 16, no. 1, pp. 105–129, 2008.

M.-L. Fang and W. Hong, “A unicity theorem for entire functions concerning differential polynomials”, Indian J. Pure Appl. Math., vol. 32, no. 9, pp. 1343–1348, 2001.

M.-L. Fang and X.-H. Hua, “Entire functions that share one value”, Nanjing Daxue Xuebao Shuxue Bannian Kan, vol. 13, no. 1, pp. 44–48, 1996.

R. G. Halburd and R. J. Korhonen, “Nevanlinna theory for the difference operator”, Ann. Acad. Sci. Fenn. Math., vol. 31, no. 2, pp. 463–478, 2006.

R. G. Halburd and R. J. Korhonen, “Difference analogue of the lemma on the logarithmic derivative with application to difference equations”, J. Math. Anal. Appl., vol. 314, no. 2, pp. 477–487, 2006.

W. K. Hayman, “Picard values of meromorphic functions and their derivatives”, Ann. of Math. (2), vol. 70, pp. 9–42, 1959.

W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Oxford: Clarendon Press, 1964.

J. Heittokangas, R. Korhonen, I. Laine and J. Rieppo, “Uniqueness of meromorphic functions sharing values with their shifts”, Complex Var. Elliptic Equ., vol. 56, no. 1–4, pp. 81–92, 2011.

J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo and J.-L. Zhang, “Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity”, J. Math. Anal. Appl., vol. 355, no. 1, pp. 352–363, 2009.

I. Lahiri, “Value distribution of certain differential polynomials”, Int. J. Math. Math. Sci., vol. 28, no. 2, pp. 83–91, 2001.

I. Lahiri, “Weighted sharing and uniqueness of meromorphic functions”, Nagoya Math. J., vol. 161, pp. 193–206, 2001.

I. Lahiri, “Weighted value sharing and uniqueness of meromorphic functions”, Complex Variables Theory Appl., vol. 46, no. 3, pp. 241–253, 2001.

I. Lahiri and A. Banerjee, “Weighted sharing of two sets”, Kyungpook Math. J., vol. 46, no. 1, pp. 79–87, 2006.

I. Lahiri and S. Dewan, “Value distribution of the product of a meromorphic function and its derivative”, Kodai Math. J., vol. 26, no. 1, pp. 95–100, 2003.

I. Laine, “Nevanlinna theory and complex differential equations”, De Gruyter Studies in Mathematics, 15, Berlin: Walter de Gruyter & Co., 1993.

I. Laine and C. C. Yang, “Value distribution of difference polynomials”, Proc. Japan Acad. Ser. A Math. Sci., vol. 83, no. 8, pp. 148–151, 2007.

X.-M. Li, W.-L. Li, H.-X. Yi and Z.-T. Wen, “Uniqueness theorems for entire functions whose difference polynomials share a meromorphic function of a smaller order”, Ann. Polon. Math., vol. 102, no. 2, pp. 111–127, 2011.

W.-L. Li and X.-M. Li, “Results on uniqueness of entire functions related to difference polynomial”, Bull. Malays. Math. Sci. Soc., vol. 39, no. 2, pp. 499–515, 2016.

W.-C. Lin and H.-X. Yi, “Uniqueness theorems for meromorphic function concerning fixed-points”, Complex Var. Theory Appl., vol. 49, no. 11, pp. 793–806, 2004.

Z. Latreuch, “On the existence of entire solutions of certain class of nonlinear difference equations”, Mediterr. J. Math., vol. 14, no. 3, paper No. 115, 16 pages, 2017.

K. Liu and L.-Z. Yang, “Value distribution of the difference operator”, Arch. Math. (Basel), vol. 92, no. 3, pp. 270–278, 2009.

K. Liu, X.-L. Liu and T.-B. Cao, “Value distributions and uniqueness of difference polynomials”, Adv. Difference Equ., Art. ID 234215, 12 pages, 2011.

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, vol. 27, no. 1, pp. 94–104, 2012.

X. Luo and W.-C. Lin, “Value sharing results for shifts of meromorphic functions”, J. Math. Anal. Appl., vol. 377, no. 2, pp. 441–449, 2011.

A. Z. Mohon‘ko, “The Nevanlinna characteristics of certain meromorphic functions”, Teor. FunkciÄ Funkcional. Anal. i Priložen, No. 14, pp. 83–87, 1971.

P. Sahoo and S. Seikh, “Uniqueness of meromorphic functions sharing a nonzero polynomial with finite weight”, Lobachevskii J. Math., vol. 34, no. 1, pp. 106–115, 2013.

P. Sahoo and S. Seikh, “Nonlinear differential polynomials sharing a small function”, Mat. Vesnik, vol. 65, no. 2, pp. 151–165, 2013.

P. Sahoo and H. Karmakar, “Results on uniqueness of entire functions whose difference polynomials share a polynomial”, J. Contemp. Math. Anal., vol. 52, no. 2, pp. 102–110, 2017.

A. Schweizer, “What is the definition of two meromorphic functions sharing a small function?”, arXiv:1705.05048v2.

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

L. Yang, Value distribution theory, Berlin: Springer-Verlag, 1993.

C.-C Yang and X.-H. Hua, “Uniqueness and value-sharing of meromorphic functions”, Ann. Acad. Sci. Fenn. Math., vol. 22, no. 2, pp. 395–406, 1997.

C.-C. Yang and I. Laine, “On analogies between nonlinear difference and differential equations”, Proc. Japan Acad. Ser. A Math. Sci., vol. 86, no. 1, pp. 10–14, 2010.

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

H.-X. Yi, “Meromorphic functions that share one or two values”, Complex Variables Theory Appl., vol. 28, no. 1, pp. 1–11, 1995.

H.-X. Yi, “Meromorphic functions that share one or two values. II”, Kodai Math. J., vol. 22, no. 2, pp. 264–272, 1999.

J.-L. Zhang, “Value distribution and shared sets of differences of meromorphic functions”, J. Math. Anal. Appl., vol. 367, no. 2, pp. 401–408, 2010.