Volume 35, Issue 3
Further Results on Meromorphic Functions and Their $n$th Order Exact Differences with Three Shared Values

Shengjiang Chen, Aizhu Xu & Xiuqing Lin

Commun. Math. Res., 35 (2019), pp. 283-288.

Published online: 2019-12

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  • Abstract

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. 

  • Keywords

meromorphic function, exact difference, uniqueness, shared value

  • AMS Subject Headings

30D35, 39A10

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

ndsycsj@126.com (Shengjiang Chen)

xuaizhu@126.com (Aizhu Xu)

  • BibTex
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  • TXT
@Article{CMR-35-283, author = {Chen , Shengjiang and Xu , Aizhu and Lin , Xiuqing}, title = {Further Results on Meromorphic Functions and Their $n$th Order Exact Differences with Three Shared Values}, journal = {Communications in Mathematical Research }, year = {2019}, volume = {35}, number = {3}, pages = {283--288}, abstract = {

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} }
TY - JOUR T1 - Further Results on Meromorphic Functions and Their $n$th Order Exact Differences with Three Shared Values AU - Chen , Shengjiang AU - Xu , Aizhu AU - Lin , Xiuqing JO - Communications in Mathematical Research VL - 3 SP - 283 EP - 288 PY - 2019 DA - 2019/12 SN - 35 DO - http://doi.org/10.13447/j.1674-5647.2019.03.09 UR - https://global-sci.org/intro/article_detail/cmr/13533.html KW - meromorphic function, exact difference, uniqueness, shared value AB -

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. 

Sheng-jiang Chen, Ai-zhu Xu & Xiu-qing Lin. (2019). Further Results on Meromorphic Functions and Their $n$th Order Exact Differences with Three Shared Values. Communications in Mathematical Research . 35 (3). 283-288. doi:10.13447/j.1674-5647.2019.03.09
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