# Meromorphic function sharing a small function with a linear differential polynomial

Mathematica Bohemica (2016)

• Volume: 141, Issue: 1, page 1-11
• ISSN: 0862-7959

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## Abstract

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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 $a=a\left(z\right)$ ($¬\equiv 0,\infty$) is a small function of $f$. If $f-a$ and $L-a$ share $0$ CM and $\left(k+1\right)\overline{N}\left(r,\infty ;f\right)+\overline{N}\left(r,0;{f}^{\text{'}}\right)+{N}_{k}\left(r,0;{f}^{\text{'}}\right)<\lambda T\left(r,{f}^{\text{'}}\right)+S\left(r,{f}^{\text{'}}\right)$ for some real constant $\lambda \in \left(0,1\right)$, then $f-a=\left(1+c/a\right)\left(L-a\right)$, where $c$ is a constant and $1+c/a¬\equiv 0$.

## How to cite

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Lahiri, Indrajit, and Sarkar, Amit. "Meromorphic function sharing a small function with a linear differential polynomial." Mathematica Bohemica 141.1 (2016): 1-11. <http://eudml.org/doc/276800>.

@article{Lahiri2016,
abstract = {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 $a = a(z)$ ($\lnot \equiv 0, \infty$) is a small function of $f$. If $f-a$ and $L-a$ share $0$ CM and $(k+1)\overline\{N\}(r, \infty ; f)+ \overline\{N\}(r, 0; f^\{\prime \})+ N\_\{k\}(r, 0; f^\{\prime \})< \lambda T(r, f^\{\prime \})+ S(r, f^\{\prime \})$ for some real constant $\lambda \in (0, 1)$, then $f-a=(1+ \{c\}/\{a\})(L-a)$, where $c$ is a constant and $1+\{c\}/\{a\} \lnot \equiv 0$.},
author = {Lahiri, Indrajit, Sarkar, Amit},
journal = {Mathematica Bohemica},
keywords = {meromorphic function; differential polynomial; small function; sharing},
language = {eng},
number = {1},
pages = {1-11},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Meromorphic function sharing a small function with a linear differential polynomial},
url = {http://eudml.org/doc/276800},
volume = {141},
year = {2016},
}

TY - JOUR
AU - Lahiri, Indrajit
AU - Sarkar, Amit
TI - Meromorphic function sharing a small function with a linear differential polynomial
JO - Mathematica Bohemica
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 141
IS - 1
SP - 1
EP - 11
AB - 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 $a = a(z)$ ($\lnot \equiv 0, \infty$) is a small function of $f$. If $f-a$ and $L-a$ share $0$ CM and $(k+1)\overline{N}(r, \infty ; f)+ \overline{N}(r, 0; f^{\prime })+ N_{k}(r, 0; f^{\prime })< \lambda T(r, f^{\prime })+ S(r, f^{\prime })$ for some real constant $\lambda \in (0, 1)$, then $f-a=(1+ {c}/{a})(L-a)$, where $c$ is a constant and $1+{c}/{a} \lnot \equiv 0$.
LA - eng
KW - meromorphic function; differential polynomial; small function; sharing
UR - http://eudml.org/doc/276800
ER -

## References

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