On periodicity of a meromorphic function when sharing two sets IM

Molla Basir Ahamed

Abstract


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.


Keywords


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

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References


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

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