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Meromorphic Functions Sharing a Nonzero Value with their Derivatives

Li, Xiao-Min;Ullah, Rahman;Piao, Da-Xiong;Yi, Hong-Xun

  • Received : 2014.07.30
  • Accepted : 2014.11.19
  • Published : 2015.03.23

Abstract

Let f be a transcendental meromorphic function of finite order in the plane such that $f^{(m)}$ has finitely many zeros for some positive integer $m{\geq}2$. Suppose that $f^{(k)}$ and f share a CM, where $k{\geq}1$ is a positive integer, $a{\neq}0$ is a finite complex value. Then f is an entire function such that $f^{(k)}-a=c(f-a)$, where $c{\neq}0$ is a nonzero constant. The results in this paper are concerning a conjecture of Bruck [5]. An example is provided to show that the results in this paper, in a sense, are the best possible.

Keywords

Meromorphic functions;Order of growth;Shared values;Uniqueness theorems

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Acknowledgement

Supported by : NSFC, NSF of Shandong Province