## Meromorphic function sharing a small function with a linear differential polynomial.(English)Zbl 1389.30136

Summary: 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 L. A. Rubel and C.-C. Yang [Complex Anal., Proc. Conf., Lexington 1976, Lect. Notes Math. 599, 101–103 (1977; Zbl 0362.30026)]. 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 $$a=a(z)$$ ($$\not\equiv 0,\infty$$) is a small function of $$f$$. If $$f-a$$ and $$L-a$$ share $$0$$ CM and $(k+1)\overline{N}(r,\infty; f)+\overline{N}(r,0;f')+N_{k}(r,0;f')<\lambda T(r,f')+S(r,f')$ for some real constant $$\lambda\in (0,1)$$, then $$f-a=(1+{c}/{a})(L-a)$$, where $$c$$ is a constant and $$1+{c}/{a}\not\equiv 0$$.

### MSC:

 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

Zbl 0362.30026
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