Uniqueness of meromorphic functions concerning fixed points
In this paper, we study a uniqueness question of meromorphic functions concerning fixed points and mainly prove the following theorem: Let $ f $ and $ g $ be two nonconstant meromorphic functions, let $ n, k $ be two positive integers with $ n > 3k+10.5-\Theta_{\min}(k+6.5) $, if $ \Theta_{\m...
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| Language: | English |
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AIMS Press
2022-09-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20221122?viewType=HTML |
| _version_ | 1817983570151473152 |
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| author | Jinyu Fan Mingliang Fang Jianbin Xiao |
| author_facet | Jinyu Fan Mingliang Fang Jianbin Xiao |
| author_sort | Jinyu Fan |
| collection | DOAJ |
| description | In this paper, we study a uniqueness question of meromorphic functions concerning fixed points and mainly prove the following theorem: Let $ f $ and $ g $ be two nonconstant meromorphic functions, let $ n, k $ be two positive integers with $ n > 3k+10.5-\Theta_{\min}(k+6.5) $, if $ \Theta_{\min}\geq \frac{2.5}{k+6.5} $, otherwise $ n > 3k+8 $, and let $ (f^{n})^{(k)} $ and $ (g^{n})^{(k)} $ share $ z $ CM, $ f $ and $ g $ share $ \infty $ IM, then one of the following two cases holds: If $ k = 1 $, then either $ f(z) = c_{1}e^{cz^{2}} $, $ g(z) = c_{2}e^{-cz^{2}} $, where $ c_{1}, c_{2} $ and $ c $ are three constants satisfying $ 4n^{2}(c_{1}c_{2})^{n}c^{2} = -1 $, or $ f = tg $ for a constant $ t $ such that $ t^{n} = 1 $; if $ k\geq2 $, then $ f = tg $ for a constant $ t $ such that $ t^{n} = 1 $. Our results extend and improve some results due to <sup>[<xref ref-type="bibr" rid="b8">8</xref>,<xref ref-type="bibr" rid="b9">9</xref>,<xref ref-type="bibr" rid="b19">19</xref>,<xref ref-type="bibr" rid="b24">24</xref>]</sup>. |
| first_indexed | 2024-04-13T23:34:44Z |
| format | Article |
| id | doaj.art-d81d13900de6408483e47f8b35995f37 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-13T23:34:44Z |
| publishDate | 2022-09-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-d81d13900de6408483e47f8b35995f372022-12-22T02:24:48ZengAIMS PressAIMS Mathematics2473-69882022-09-01712204902050910.3934/math.20221122Uniqueness of meromorphic functions concerning fixed pointsJinyu Fan0Mingliang Fang 1Jianbin Xiao2Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, ChinaDepartment of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, ChinaDepartment of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, ChinaIn this paper, we study a uniqueness question of meromorphic functions concerning fixed points and mainly prove the following theorem: Let $ f $ and $ g $ be two nonconstant meromorphic functions, let $ n, k $ be two positive integers with $ n > 3k+10.5-\Theta_{\min}(k+6.5) $, if $ \Theta_{\min}\geq \frac{2.5}{k+6.5} $, otherwise $ n > 3k+8 $, and let $ (f^{n})^{(k)} $ and $ (g^{n})^{(k)} $ share $ z $ CM, $ f $ and $ g $ share $ \infty $ IM, then one of the following two cases holds: If $ k = 1 $, then either $ f(z) = c_{1}e^{cz^{2}} $, $ g(z) = c_{2}e^{-cz^{2}} $, where $ c_{1}, c_{2} $ and $ c $ are three constants satisfying $ 4n^{2}(c_{1}c_{2})^{n}c^{2} = -1 $, or $ f = tg $ for a constant $ t $ such that $ t^{n} = 1 $; if $ k\geq2 $, then $ f = tg $ for a constant $ t $ such that $ t^{n} = 1 $. Our results extend and improve some results due to <sup>[<xref ref-type="bibr" rid="b8">8</xref>,<xref ref-type="bibr" rid="b9">9</xref>,<xref ref-type="bibr" rid="b19">19</xref>,<xref ref-type="bibr" rid="b24">24</xref>]</sup>.https://www.aimspress.com/article/doi/10.3934/math.20221122?viewType=HTMLmeromorphic functionentire functionunicityfixed point |
| spellingShingle | Jinyu Fan Mingliang Fang Jianbin Xiao Uniqueness of meromorphic functions concerning fixed points AIMS Mathematics meromorphic function entire function unicity fixed point |
| title | Uniqueness of meromorphic functions concerning fixed points |
| title_full | Uniqueness of meromorphic functions concerning fixed points |
| title_fullStr | Uniqueness of meromorphic functions concerning fixed points |
| title_full_unstemmed | Uniqueness of meromorphic functions concerning fixed points |
| title_short | Uniqueness of meromorphic functions concerning fixed points |
| title_sort | uniqueness of meromorphic functions concerning fixed points |
| topic | meromorphic function entire function unicity fixed point |
| url | https://www.aimspress.com/article/doi/10.3934/math.20221122?viewType=HTML |
| work_keys_str_mv | AT jinyufan uniquenessofmeromorphicfunctionsconcerningfixedpoints AT mingliangfang uniquenessofmeromorphicfunctionsconcerningfixedpoints AT jianbinxiao uniquenessofmeromorphicfunctionsconcerningfixedpoints |