On periodicity of a meromorphic function when sharing two sets IM

Molla Basir Ahamed


In this paper, we have investigated the sufficient conditions for periodicity of meromorphic functions and obtained two results directly improving
the result of \emph{Bhoosnurmath-Kabbur} \cite{Bho & Kab-2013}, \emph{Qi-Dou-Yang} \cite{Qi & Dou & Yan-ADE-2012} and \emph{Zhang} \cite{Zha-JMMA-2010}. Let $\mathcal{S}_{1}=\left\{z:\displaystyle\int_{0}^{z-a}(t-a)^n(t-b)^4dt+1=0\right\}$ and $\mathcal{S}_{2}=\bigg\{a,b\bigg\}$, where $n\geq 4(n\geq 3)$ be an integer.\emph{ Let $f(z)$ be a non-constant meromorphic (entire) function satisfying $\ol E_{f(z)}(\mathcal{S}_j)=\ol E_{f(z+c)}(\mathcal{S}_j), (j=1,\;2)$
then $f(z)\equiv f(z+c)$.} We have exhibited some examples to show that, it is not necessary that the meromorphic function should be of finite order and also to show that the sets considered in this paper simply can't be replace by some arbitrary sets. At the last section, we have posed an open question for the future research.


Meromorphic function, shared sets, finite and infinite order, shift operator, periodicity

Full Text:



Banerjee, A., emph{Uniqueness of meromorphic functions sharing two sets with finite weight}, Portugal.

Math. J., textbf{{65}}(2008), no. 1, 81–-93.

Banerjee, A., emph{Uniqueness of meromorphic functions sharing two sets with finite weight II}, Tamkang.

Math. J., textbf{{41}}(2010), no. 4, 379-–392.

Bhoosnurmath, S.S., Dyavanal, R.S., {em Uniqueness of meromorphic functions sharing a set}, Bull. Math. Anal. Appl., textbf{{ 3}}(2011), no. 3, 200--208.

Bhoosnurmath, S.S., Kabbur, S.R., {em Value distribution and uniqueness theorems for difference of entire and meromorphic functions}, Int. J. Anal. Appl., textbf{{ 2}}(2013), no. 2, 124--136.

Chen, B., Chen, Z., {em Meromorphic functions sharing two sets with its difference operator}, Bull. Malays.

Math. Soc., textbf{{35}}(2012), no. 3, 765-–774.

Chen, B., Chen, Z., Li, S., {em Uniqueness of difference operators of meromorphic functions}, J. Inequal. Appl., textbf{{48}}(2012), 1--19. (doi:10.1186/1029-242X-2012-48)

Fang, M., Lahiri, I., emph{Unique range set for certain meromorphic functions,} Indian J. Math., textbf{{45}}(2003), no. 2, 141--150.

Fujimoto, H., {em On uniqueness polynomials for meromorphic functions }, Nagoya Math. J., textbf{{170}}(2003), 33--46.

Goldberg, A., Ostrovskii, I., emph{Value Distribution of Meromorphic Functions}, Transl. Math. Monogr., vol. 236, American Mathematical Society, Providence, RI, 2008, translated from the 1970 Russian original by Mikhail Ostrovskii, with an appendix by Alexandre Eremenko and James K. Langley.

Gross, F., emph{Factorization of meromorphic functions and some open problems}, Proc. Conf. Univ. Kentucky, Leixngton, Ky(1976); Complex Analysis, Lecture Notes in Math., textbf{599}(1977), 51--69, Springer Verlag.

Hayman, W.K., {em Meromorphic functions}, The Clarendon Press, Oxford (1964).

Laine, I., emph{Nevanlinna Theory and Complex Differential Equations}, Walter de Gruyter, Berlin, 1993.

Mokhon’ko, A.Z., {em On the Nevanlinna characteristics of some meromorphic functions}, Theo. Funct. Funct. Anal. Appl., Izd-vo Khar’kovsk Un-ta, textbf{{ 14}}(1971), 83-–87.

Nevanlinna, R., emph{Le théorème de Picard–Borel et la théorie des fonctions méromorphes, Gauthiers–Villars}, Paris, 1929.

Qi, X.G., Dou, J., Yang, L.Z., {em Uniqueness and value distribution for difference operator of meromorphic function}, Adv. Diff. Equn., textbf{{32}} (2012), 1--9. (doi.org/10.1186/1687-1847-2012-32).

Yang, C.C., Yi, H.X., textit{Uniqueness Theory of Meromorphic Functions}, Math. Appl., textbf{557}(2003) Kluwer Academic Publishers Group, Dordrecht.

Yi, H.X. Lin, W.C., emph{Uniqueness of meromorphic functions and a question of Gross}, Kyungpook Math. J., textbf{{46}}(2006), 437--444.

Zhang, J.L., {em Value distribution and sets of difference of meromorphic functions}, J. Math. Anal. Appl., textbf{{367}}(2010), no. 2, 401--408.

Zhang, J. Xu, Y., emph{Meromorphic functions sharing two sets}, Appl. Math. Lett., textbf{{21}}(2008), 471--476.

DOI: http://dx.doi.org/10.24193/subbmath.2019.4.05


  • There are currently no refbacks.