## Uniqueness of meromorphic functions sharing a meromorphic function of a smaller order with their derivatives.(English)Zbl 1201.30034

Five uniqueness theorems for meromorphic functions and their derivatives are obtained. One of the results is the following: Let $$f(z)$$ be a meromorphic function of finite order, and let $$a(z)$$ be a meromorphic function satisfying $$\sigma(a)<\sigma(f)$$. Assume that $$f(z)$$ and $$a(z)$$ have finitely many poles and have no common poles. If $$f(z)- a(z)$$ and $$f^{(k)}(z)- a(z)$$ share $$0$$ CM, where $$k\geq 1$$ is a positive integer, then $$f^{(k)}(z)- a(z)= c\big(f(z)- a(z)\big)$$ for some constant $$c$$. This result is a generalization of [J. M. Chang and Y. Z. Zhu, J. Math. Anal. Appl. 351, No. 1, 491–496 (2009; Zbl 1162.30021)]. The tools of the proof are from Nevanlinna theory and Wiman-Valiron theory.

### MSC:

 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory 30D30 Meromorphic functions of one complex variable (general theory)

Zbl 1162.30021
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