Uniqueness theorems of meromorphic functions of a certain form
Bull. Korean Math. Soc. 2009 Vol. 46, No. 6, 1079-1089
Published online November 1, 2009
Junfeng Xu, Qi Han, and Jilong Zhang
Wuyi University, University of Houston, and Beihang University
Abstract : In this paper, we shall show that for any entire function $f$, the function of the form $f^m(f^n-1)f^{\prime}$ has no non-zero finite Picard value for all positive integers $m$, $n\in \mathbb{N}$ possibly except for the special case $m=n=1$. Furthermore, we shall also show that for any two non-constant meromorphic functions $f$ and $g$, if $f^m(f^n-1)f^{\prime}$ and $g^m(g^n-1)g^{\prime}$ share the value 1 weakly, then $f\equiv g$ provided that $m$ and $n$ satisfy some conditions. In particular, if $f$ and $g$ are entire, then the restrictions on $m$ and $n$ could be greatly reduced.
Keywords : entire function, meromorphic function, Picard value
MSC numbers : 30D35, 30D20, 30D30
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