# 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 $1+c/a¬\equiv 0$.

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