Institute of Mathematics

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 $a = a(z)$ ($\not\equiv0, \infty$) 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')+ 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 $1+c/a \not\equiv0$.