Kyungpook Mathematical Journal 2015; 55(1): 137-147
Published online March 23, 2015
Copyright © Kyungpook Mathematical Journal.
Meromorphic Functions Sharing a Nonzero Value with their Derivatives
Xiao-Min Li1, Rahman Ullah1, Da-Xiong Piao1 and Hong-Xun Yi2
1School of Mathematical Sciences, Ocean University Of China, Qingdao, Shandong 266100, P. R. China
2Department of Mathematics, Shandong University, Jinan, Shandong 251000, P. R. China
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 $mgeq 2.$ Suppose that $f^{(k)}$ and $f$ share $a$ CM, where $kgeq 1$ is a positive integer, $a
eq 0$ is a finite complex value. Then $f$ is an entire function such that $f^{(k)}-a=c(f-a),$ where $c
eq 0$ is a nonzero constant. The results in this paper are concerning a conjecture of Br"{u}ck [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