Computational Methods and Function Theory 7 (2007), No. 1, 141--149
Copyright Heldermann Verlag 2007
Indrajit Lahiri
indr9431@dataone.in , University of Kalyani, Department of Mathematics, West Bengal 741235, India.
Kit-Wing Yu
kitwing@hotmail.com , Rm. 205, Kwai Shun Hse., Kwai Fong Est., Kwai Chung, Hong Kong, China.
Let $p$ be a positive integer not less than $2$. It is shown that a necessary condition for the generalized Fermat type functional equation $$ \sum_{j=1}^pa_j(z){f_j}^{k_j}(z)\equiv 1 $$ having non-constant meromorphic solutions $f_1,f_2,\ldots,f_p$ is $$ \sum_{j=1}^p\frac{1}{k_j} \ge \frac{1}{(p-1)+A_p}, $$ where $A_2=1/2$, $A_p=(2p-3)/3$ if $p=3,4,5$, $A_p=(2p+1-2\sqrt{2p})/2$ if $p\geq6$ and $T(r,a_j)=\Landauo(T(r,f_j))$, $1 \le j \le p$, as $r \to +\infty$, $r \not\in E$ and $E$ is a set of finite linear measure. This improves the result of Yu and Yang \cite{Yu} in 2002. Next we discuss a question of Hayman \cite{Hayman2} and give a partial answer to it.
Keywords: Differential equations, generalized Fermat type functional equations, linearly independent, meromorphic functions, Nevanlinna theory, uniqueness theory of meromorphic functions.
MSC 2000: 30D05, 30D30, 30D35.
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