Meromorphic functions sharing three values with their shift
UDC 517.5 We discuss the  problem of uniqueness of a meromorphic function $f(z),$ which shares $a_1(z)$, $a_2(z),$ and $a_3(z)$ CM with its shift $f(z+c)$, where $a_1(z)$, $a_2(z),$ and $a_3(z)$ are three $c$-periodic distinct small functions of $f(z)$ and $c\i...
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| Date: | 2024 |
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| Language: | English |
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Institute of Mathematics, NAS of Ukraine
2024
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7502 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512677884854272 |
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| author | Majumder, Sujoy Das, Pradip Majumder, Sujoy Majumder, Sujoy Das, Pradip |
| author_facet | Majumder, Sujoy Das, Pradip Majumder, Sujoy Majumder, Sujoy Das, Pradip |
| author_sort | Majumder, Sujoy |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2024-07-15T03:05:05Z |
| description | UDC 517.5
We discuss the  problem of uniqueness of a meromorphic function $f(z),$ which shares $a_1(z)$, $a_2(z),$ and $a_3(z)$ CM with its shift $f(z+c)$, where $a_1(z)$, $a_2(z),$ and $a_3(z)$ are three $c$-periodic distinct small functions of $f(z)$ and $c\in\mathbb{C}\setminus\{0\}$. The obtained result improves the recent result of Heittokangas et al. [Complex Var. and Elliptic Equat., 56, No. 1–4, 81–92 (2011)]  by dropping the assumption about the order of $f(z)$.  In addition, we introduce a way of characterizing elliptic functions in terms of meromorphic functions  sharing values with two of their shifts.  Moreover, we show by  a number of illustrating examples that  our results are,  in certain senses, best possible. |
| doi_str_mv | 10.3842/umzh.v76i5.7502 |
| first_indexed | 2026-03-24T03:32:36Z |
| format | Article |
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| id | umjimathkievua-article-7502 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:32:36Z |
| publishDate | 2024 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-75022024-07-15T03:05:05Z Meromorphic functions sharing three values with their shift Meromorphic functions sharing three values with their shift Majumder, Sujoy Das, Pradip Majumder, Sujoy Majumder, Sujoy Das, Pradip meromorphic function; uniqueness theory; shared values; Nevanlinna theory; shift; difference. UDC 517.5 We discuss the  problem of uniqueness of a meromorphic function $f(z),$ which shares $a_1(z)$, $a_2(z),$ and $a_3(z)$ CM with its shift $f(z+c)$, where $a_1(z)$, $a_2(z),$ and $a_3(z)$ are three $c$-periodic distinct small functions of $f(z)$ and $c\in\mathbb{C}\setminus\{0\}$. The obtained result improves the recent result of Heittokangas et al. [Complex Var. and Elliptic Equat., 56, No. 1–4, 81–92 (2011)]  by dropping the assumption about the order of $f(z)$.  In addition, we introduce a way of characterizing elliptic functions in terms of meromorphic functions  sharing values with two of their shifts.  Moreover, we show by  a number of illustrating examples that  our results are,  in certain senses, best possible. УДК 517.5 Мероморфні функції, що поділяють три значення з їхнім зсувом  Обговорено проблему єдиності мероморфної функції $f(z)$, що має спільні $a_1(z)$, $a_2(z)$ і $a_3(z)$ CM    зі своїм зсувом $f(z+c )$, де $a_1(z)$, $a_2(z)$ і $a_3(z)$ — три $c$-періодичні різні малі функції від $f(z)$ та   $c\in\mathbb{C }\setminus\{0\}$.  Отриманий результат покращує останній результат Heittokangas та ін. [Complex Var. and Elliptic Equat., 56, No. 1–4, 81–92 (2011)] , оскільки  відкинуто припущення про порядок $f(z)$. Крім того, запропоновано спосіб характеризації еліптичних функцій у термінах мероморфних функцій, що поділяють значення з двома своїми зсувами. Насамкінець, на основі розгляду  низки ілюстративних прикладів, показано, що одержані результати є найкращими у певному сенсі. Institute of Mathematics, NAS of Ukraine 2024-07-03 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7502 10.3842/umzh.v76i5.7502 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 6 (2024); 877–889 Український математичний журнал; Том 76 № 6 (2024); 877–889 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7502/10034 Copyright (c) 2024 Sujoy Majumder, Pradip Das |
| spellingShingle | Majumder, Sujoy Das, Pradip Majumder, Sujoy Majumder, Sujoy Das, Pradip Meromorphic functions sharing three values with their shift |
| title | Meromorphic functions sharing three values with their shift |
| title_alt | Meromorphic functions sharing three values with their shift |
| title_full | Meromorphic functions sharing three values with their shift |
| title_fullStr | Meromorphic functions sharing three values with their shift |
| title_full_unstemmed | Meromorphic functions sharing three values with their shift |
| title_short | Meromorphic functions sharing three values with their shift |
| title_sort | meromorphic functions sharing three values with their shift |
| topic_facet | meromorphic function uniqueness theory shared values Nevanlinna theory shift difference. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7502 |
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