Meromorphic function sharing a small function with a linear differential polynomial
The problem of uniqueness of an entire or a meromorphic function when it shares a value or a small function with its derivative became popular among the researchers after the work of Rubel and Yang (1977). Several authors extended the problem to higher order derivatives. Since a linear differential...
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| Format: | Article |
| Language: | English |
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Institute of Mathematics of the Czech Academy of Science
2016-04-01
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| Series: | Mathematica Bohemica |
| Subjects: | |
| Online Access: | http://mb.math.cas.cz/full/141/1/mb141_1_1.pdf |
| _version_ | 1818877412226629632 |
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| author | Indrajit Lahiri Amit Sarkar |
| author_facet | Indrajit Lahiri Amit Sarkar |
| author_sort | Indrajit Lahiri |
| collection | DOAJ |
| description | The problem of uniqueness of an entire or a meromorphic function when it shares a value or a small function with its derivative became popular among the researchers after the work of Rubel and Yang (1977). Several authors extended the problem to higher order derivatives. Since a linear differential polynomial is a natural extension of a derivative, in the paper we study the uniqueness of a meromorphic function that shares one small function CM with a linear differential polynomial, and prove the following result: Let $f$ be a nonconstant meromorphic function and $L$ a nonconstant linear differential polynomial generated by $f$. Suppose that $a = a(z)$ ($\not\equiv0, \infty$) is a small function of $f$. If $f-a$ and $L-a$ share $0$ CM and
(k+1)øverline N(r, \infty; f)+ øverline N(r, 0; f')+ N_k(r, 0; f')< \lambda T(r, f')+ S(r, f')
for some real constant $\lambda\in(0, 1)$, then $ f-a=(1+ c/a)(L-a)$, where $c$ is a constant and $1+c/a \not\equiv0$. |
| first_indexed | 2024-12-19T13:57:52Z |
| format | Article |
| id | doaj.art-267828628132412598cb7f8c75843d66 |
| institution | Directory Open Access Journal |
| issn | 0862-7959 2464-7136 |
| language | English |
| last_indexed | 2024-12-19T13:57:52Z |
| publishDate | 2016-04-01 |
| publisher | Institute of Mathematics of the Czech Academy of Science |
| record_format | Article |
| series | Mathematica Bohemica |
| spelling | doaj.art-267828628132412598cb7f8c75843d662022-12-21T20:18:32ZengInstitute of Mathematics of the Czech Academy of ScienceMathematica Bohemica0862-79592464-71362016-04-01141111110.21136/MB.2016.1MB.2016.1Meromorphic function sharing a small function with a linear differential polynomialIndrajit LahiriAmit SarkarThe problem of uniqueness of an entire or a meromorphic function when it shares a value or a small function with its derivative became popular among the researchers after the work of Rubel and Yang (1977). Several authors extended the problem to higher order derivatives. Since a linear differential polynomial is a natural extension of a derivative, in the paper we study the uniqueness of a meromorphic function that shares one small function CM with a linear differential polynomial, and prove the following result: Let $f$ be a nonconstant meromorphic function and $L$ a nonconstant linear differential polynomial generated by $f$. Suppose that $a = a(z)$ ($\not\equiv0, \infty$) is a small function of $f$. If $f-a$ and $L-a$ share $0$ CM and (k+1)øverline N(r, \infty; f)+ øverline N(r, 0; f')+ N_k(r, 0; f')< \lambda T(r, f')+ S(r, f') for some real constant $\lambda\in(0, 1)$, then $ f-a=(1+ c/a)(L-a)$, where $c$ is a constant and $1+c/a \not\equiv0$.http://mb.math.cas.cz/full/141/1/mb141_1_1.pdf meromorphic function differential polynomial small function sharing |
| spellingShingle | Indrajit Lahiri Amit Sarkar Meromorphic function sharing a small function with a linear differential polynomial Mathematica Bohemica meromorphic function differential polynomial small function sharing |
| title | Meromorphic function sharing a small function with a linear differential polynomial |
| title_full | Meromorphic function sharing a small function with a linear differential polynomial |
| title_fullStr | Meromorphic function sharing a small function with a linear differential polynomial |
| title_full_unstemmed | Meromorphic function sharing a small function with a linear differential polynomial |
| title_short | Meromorphic function sharing a small function with a linear differential polynomial |
| title_sort | meromorphic function sharing a small function with a linear differential polynomial |
| topic | meromorphic function differential polynomial small function sharing |
| url | http://mb.math.cas.cz/full/141/1/mb141_1_1.pdf |
| work_keys_str_mv | AT indrajitlahiri meromorphicfunctionsharingasmallfunctionwithalineardifferentialpolynomial AT amitsarkar meromorphicfunctionsharingasmallfunctionwithalineardifferentialpolynomial |