Bull. Korean Math. Soc. 2009; 46(6): 1079-1089
Printed November 1, 2009
https://doi.org/10.4134/BKMS.2009.46.6.1079
Copyright © The Korean Mathematical Society.
Junfeng Xu, Qi Han, and Jilong Zhang
Wuyi University, University of Houston, and Beihang University
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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