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Zbl 0924.39017
Bergweiler, Walter; Ishizaki, Katsuya; Yanagihara, Niro
Meromorphic solutions of some functional equations.
(English)
[J] Methods Appl. Anal. 5, No.3, 248-258 (1998); correction ibid. 6, No.4, 617-618 (1999). ISSN 1073-2772

Authors' abstract: It is shown that transcendental meromorphic solutions $f(z)$ of the functional equation $$\sum^n_{j=0} a_j(z)f(c^jz)=Q(z),$$ where $0<| c|<1$ is a complex number and $a_j(z)$, $j=0,1,\dots,n$ and $Q(z)$ are rational functions with $a_0(z)\not\equiv 0$, $a_n(z)\equiv 1$, satisfy $$T(r,f)=O\bigl((\log r)^2\bigr) \quad\text {and}\quad (\log r)^2=O\bigl(T(r,f)\bigr),$$ where $T(r,f)$ is the characteristic function of $f(z)$. Moreover, in the case $n=2$ and $Q(z)\equiv 0$, necessary and sufficient conditions for the existence of solutions are given.
[B.Crstici (Timişoara)]
MSC 2000:
*39B32 Functional equations for complex functions
30D05 Functional equations in the complex domain

Keywords: transcendental meromorphic solutions; functional equation

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