Meromorphic functions sharing a nonzero polynomial CM
Bull. Korean Math. Soc. 2010 Vol. 47, No. 2, 319-339
https://doi.org/10.4134/BKMS.2010.47.2.319
Printed March 1, 2010
Xiao-Min Li and Ling Gao
Ocean University of China and Ocean University of China
Abstract : In this paper, we prove that if $f^nf'-P$ and $g^ng'-P$ share $0$ CM, where $f$ and $g$ are two distinct transcendental meromorphic functions, $n\geq 11$ is a positive integer, and $P$ is a nonzero polynomial such that its degree $\gamma_P \leq 11$, then either $f=c_1e^{cQ}$ and $g=c_2e^{-cQ},$ where $c_1,$ $c_2$ and $c$ are three nonzero complex numbers satisfying $(c_1c_2)^{n+1}c^2=-1,$ $Q$ is a polynomial such that $Q=\int_0^zP(\eta)d\eta,$ or $f=tg$ for a complex number $t$ such that $t^{n+1}=1.$ The results in this paper improve those given by M. L. Fang and H. L. Qiu, C. C. Yang and X. H. Hua, and other authors.
Keywords : meromorphic functions, shared values, differential polynomials, uniqueness theorems
MSC numbers : 30D35, 30D30
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