Commun. Math. Res., 35 (2019), pp. 283-288.
Published online: 2019-12
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Let $E(a,\,f)$ be the set of $a$-points of a meromorphic function $f(z)$ counting multiplicities. We prove that if a transcendental meromorphic function $f(z)$ of hyper order strictly less than 1 and its $n$th exact difference $\Delta_c^nf(z)$ satisfy $E(1,\,f)=E(1,\,\Delta_c^nf)$, $E(0,\,f)\subset E(0,\,\Delta_c^nf)$ and $E(\infty,\,f)\supset E(\infty,\,\Delta_c^nf)$, then $\Delta_c^nf(z)\equiv f(z)$. This result improves a more recent theorem due to Gao et al. (Gao Z, Kornonen R, Zhang J, Zhang Y. Uniqueness of meromorphic functions sharing values with their $n$th order exact differences. Analysis Math., 2018, https://doi.org/10.1007/s10476018-0605-2) by using a simple method.
}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2019.03.09}, url = {http://global-sci.org/intro/article_detail/cmr/13533.html} }Let $E(a,\,f)$ be the set of $a$-points of a meromorphic function $f(z)$ counting multiplicities. We prove that if a transcendental meromorphic function $f(z)$ of hyper order strictly less than 1 and its $n$th exact difference $\Delta_c^nf(z)$ satisfy $E(1,\,f)=E(1,\,\Delta_c^nf)$, $E(0,\,f)\subset E(0,\,\Delta_c^nf)$ and $E(\infty,\,f)\supset E(\infty,\,\Delta_c^nf)$, then $\Delta_c^nf(z)\equiv f(z)$. This result improves a more recent theorem due to Gao et al. (Gao Z, Kornonen R, Zhang J, Zhang Y. Uniqueness of meromorphic functions sharing values with their $n$th order exact differences. Analysis Math., 2018, https://doi.org/10.1007/s10476018-0605-2) by using a simple method.